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MODAL ANALYSIS OF CONTROL VALVE LIFTING BAR ASSEMBLY
______________________
A Thesis
Presented to
the Faculty of the Graduate School
Tennessee Technological University
by
Rajendra Prasad Vavilala
_________________
In Partial Fulfillment
of the Requirements for the Degree
MASTER OF SCIENCE
Mechanical Engineering
____________
August 2000
ii
CERTIFICATE OF APPROVAL OF THESIS
MODAL ANALYSIS OF CONTROL VALVE LIFTING BAR ASSEMBLY
by
Rajendra Prasad Vavilala
Graduate Advisory Committee:
_________________________ ___________ Chairperson date
_________________________ ___________ Member date
_________________________ ___________ Member date
Approved for the Faculty:
______________________________ Dean of Graduate Studies
______________________________ Date
iii
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment of the requirements for a Master of
Science degree at Tennessee Technological University, I agree that the University
Library shall make it available to borrowers under rules of the Library. Brief quotations
from this thesis are allowable without special permission, provided that accurate
acknowledgement of the source is made.
Permission for extensive quotation from or reproduction of this thesis may be
granted by my major professor when the proposed use of the material is for scholarly
purposes. Any copying or use of the material in this thesis for financial gain shall not be
allowed without my written permission.
Signature ____________________________
Date________________________________
iv
ACKNOWLEDGMENTS
The author expresses his most sincere thanks and appreciation to his mentor and
chairman of his Graduate Advisory Committee, Dr. Christopher D. Wilson. The author is
forever grateful for his patience, assistance, guidance, and teachings.
The author also thanks the other members of his Graduate Advisory Committee,
Dr. Sally Pardue and Dr. Glenn Cunningham for giving their time and effort as the
members of the committee.
The author thanks Mike Steakley and Tennesse Valley Authority for providing
funding for this research. The author also thanks the Mechanical Engineering Department
and the Center for Electric Power for providing financial and personal support. The
author’s achievements during the last two years would have been impossible without
their assistance. Thanks are extended to Dr. J. Richard Houghton for guiding the author
in the experimental part of this thesis.
v
TABLE OF CONTENTS
Page
LIST OF TABLES......................................................................................................... x
LIST OF FIGURES ....................................................................................................... xii
LIST OF SYMBOLS AND ACRONYMS.................................................................... xvii
Chapter
1.INTRODUCTION ....................................................................................................
........................................................................................................................... 1
Problem Statement ..................................................................................... 1
Thesis Outline............................................................................................. 6
2.TECHNICAL BACKGROUND...............................................................................
........................................................................................................................... 8
Signal Analysis........................................................................................... 8
Random Signal Data.......................................................................... 9
Frequency Domain Data.................................................................... 12
Fourier Series ........................................................................... 12
Fourier Transforms................................................................... 13
Finite Fourier Transforms ........................................................ 13
Discrete Fourier Transforms .................................................... 14
Spectral Functions .................................................................... 15
Distributed Systems........................................................................... 17
vi
Finite Element Method............................................................................... 18
Fatigue Analysis ......................................................................................... 22
vii
Chapter Page
3.VISUAL INSPECTION OF WEAR DAMAGE ......................................................
......................................................................................................................... 28
4.ANALYTICAL AND NUMERICAL PROGRAM..................................................
......................................................................................................................... 34
Modal Analysis........................................................................................... 34
Model Preparation ............................................................................. 35
Solution ............................................................................................. 38
Examination of Results ..................................................................... 39
Mode Shape Estimation..................................................................... 39
Fatigue Analysis ......................................................................................... 40
5.EXPERIMENTAL PROGRAM ...............................................................................
......................................................................................................................... 48
Preliminary Measurements of Disassembled Lifting Bar .......................... 48
Experimental Determination of Natural Frequencies................................. 49
Selection of Transducers ............................................................................ 49
Accelerometer ................................................................................... 50
Sensitivity................................................................................. 50
Frequency bandwidth ............................................................... 50
Environmental integrity............................................................ 51
Acoustic Emitter Detector ................................................................. 52
Laser Beam Vibrometer .................................................................... 53
viii
Selection of Proper Location...................................................................... 54
Experimental Setup .................................................................................... 55
Data Signal Generation .............................................................................. 57
Chapter Page
Analysis of Experimental Data................................................................... 58
6.RESULTS AND DISCUSSION...............................................................................
......................................................................................................................... 61
Numerical Results ...................................................................................... 61
Experimental Results.................................................................................. 68
Preliminary Measurement Results at the Allen Steam Plant............. 68
Measurements at the Kingston Steam Plant ...................................... 70
Axial response .......................................................................... 70
Transverse response ................................................................. 73
Comparison of Natural Frequencies........................................................... 76
Fatigue Analysis Results ............................................................................ 78
7.CONCLUSIONS AND RECOMMENDATIONS ...................................................
......................................................................................................................... 81
REFERENCES .............................................................................................................. 84
APPENDICES
A. MATLAB PROGRAM TO TRANSFORM TIME DOMAIN DATA INTO
FREQUENCY DOMAIN DATA...................................................................... 88
B. SAMPLE DATA FILE ...................................................................................... 94
ix
C. MATLAB PROGRAM TO CALCULATE THE VARIATION IN LENGTH
OF LIFTING ROD FOR DIFFERENT POWER OUTPU LEVELS................ 95
D. MATLAB PROGRAM TO GENERATE S-N DIAGRAM OF
REFRACTALOY 26 ......................................................................................... 96
VITA.............................................................................................................................. 97
x
LIST OF TABLES
Table Page
1.1. Composition of Refractaloy 26 [3]............................................................... 5
1.2. Physical and Mechanical Properties of Refractaloy 26 at 1000 ºF [3].......... 5
4.1. Variation of Lifting Rod Length with the Percentage Output....................... 37
5.1. General Properties of the Accelerometer ...................................................... 51
5.2. General Properties of the Acoustic Emitter Detector.................................... 52
5.3. General Properties of Laser Beam Vibrometer ............................................. 54
6.1. Natural Frequencies Obtained from Modal Analysis of Control Valve
Lifting Bar for 0 Percent Output of Kingston Steam Power Plant................ 62
6.2. Natural Frequencies Obtained from Modal Analysis of Control Valve
Lifting Bar for 42 Percent Output of Kingston Steam Power Plant.............. 62
6.3. Natural Frequencies Obtained from Modal Analysis of Control Valve
Lifting Bar for 48 Percent Output of Kingston Steam Power Plant.............. 62
6.4. Natural Frequencies Obtained from Modal Analysis of Control Valve
Lifting Bar for 64 Percent Output of Steam Kingston Power Plant.............. 63
6.5. Natural Frequencies Obtained from Modal Analysis of Control Valve
Lifting Bar for 100 Percent Output of Kingston Steam Power Plant............ 63
6.6. Impact Transverse Natural Frequencies (Precision: +/-5 Hz) when
Accelerometers are Mounted on the Output Side of Lifting Rod ................. 69
xi
Table Page
6.7. Impact Transverse Natural Frequencies (Precision: +/-5hz) when
Accelerometers are Mounted on the Governor Side of Lifting Rod ............. 69
6.8. Comparison of Natural Frequencies Obtained from Numerical and
Experimental Results for Unit 1 of Kingston Steam Power Plant for 64%
Output............................................................................................................ 77
7.1. Natural Frequencies of the Modified Lifting Bar Assembly......................... 83
xii
LIST OF FIGURES
Figure Page
1.1. Steam Chest with Lifting Rod (Part 39) Identified in Section A-A
(Drawing taken from Westinghouse I.L.1250-602) [1] ................................ 2
1.2. Detailed Drawing of the Lifting Bar Assembly of Allen Steam Power
Plant used for Finite Element Analysis (adapted from [2])........................... 3
1.3. Detailed Drawing of the Lifting Bar Assembly of Kingston Steam Power
Plant used for Finite Element Analysis (adapted from [2])........................... 4
2.1. Mass, Spring, and Damper Single Degrees of Freedom System and its
Free Body Diagram ....................................................................................... 9
2.2. Ensemble of Time History Records Defining Random Process xi(t)
(adapted from [4]) ......................................................................................... 10
2.3. Ten-Node Tetrahedral Solid Element Used to Discretize the Three-
Dimensional Structure [11] ........................................................................... 19
2.4. Stress Varying in Sinusoidal Fashion Showing Mean and Alternating
Stress Experienced by a Structure ................................................................. 23
2.5. Alternating Stress at Failure versus Cycles to Failure .................................. 25
3.1. Large Half Moon Shaped Compression Buildup from High Velocity
Momentum of the Valve................................................................................ 29
3.2. Typical Wear on the Underneath Side of the Supporting Hole Resulted
from High Velocity Momentum of the Valve ............................................... 29
xiii
Figure Page
3.3. Metallic Peen Type Deformation Observed at the Interface of the Lifting
Rod Connection with the Lifting Bar ............................................................ 30
3.4. Metallic Peen Type Deformation Observed at the Interface of the Lifting
Rod Connection with the Lifting Bar ............................................................ 30
3.5. Wear on the Lifting Bar at the Twist Lock Base Connection of the Control
Valve Lifting Rod.......................................................................................... 32
3.6. Control Valve in the Lifting Bar Assembly Showing Wear in All
Directions on the Conical Seat in Equal Amount.......................................... 32
3.7. Wear on the Lifting Rod at the End where there is a Significant Radial
Force breaking through the Normal Steam Lubricant................................... 33
3.8. Wear on the Lifting Rod at the End where there is a Significant Radial
Force breaking Through the Normal Steam Lubricant ................................. 33
4.1. Geometric Model of Control Valve Lifting Bar Assembly of TVA
Kingston Power Plant.................................................................................... 36
4.2. Element Plot of Control Valve Lifting Bar Assembly of TVA Kingston
Power Plant.................................................................................................... 36
4.3. Element Plot with Boundary Conditions of the Model in Working
Orientation..................................................................................................... 38
4.4. Assumed True Stress-Strain Diagram of Refractaloy 26 for Elastic-Plastic
Stress Analysis in ANSYS ............................................................................ 41
4.5. S-N Diagram of Refractaloy 26 Generated Using MATLAB....................... 42
xiv
Figure Page
4.6. Pressure Loading Applied on One Side of the Interface of Control Valve
and the Lifting Bar ........................................................................................ 43
4.7. Force Loading Applied on One Side of the Interface of Control Valve
Resting on the Lifting Bar to Simulate the Rubbing Action ......................... 44
4.8. Pressure Loading Applied on the Front Surface of the Lifting Bar to
Simulate the Impact of Steam ....................................................................... 45
4.9. Pressure Loading and the Symmetric Constraints Applied on the Lifting
Bar Assembly ................................................................................................ 46
5.1. Experimental Setup to obtain the Vibration Response of Lifting Bar
Assembly when Lifting Bar is Out Side of the Steam Chest ........................ 48
5.2. Schematic Representation of Measurement Directions for Laser Beam
Vibrometer..................................................................................................... 53
5.3. Experimental Setup made to obtain the Vibration Response of Lifting Bar
Assembly....................................................................................................... 55
5.4. Experimental Set Up to Measure the Response of Lifting Bar Assembly at
Right Angles to the Door Opening Direction................................................ 56
6.1. Schematic Representation of Mode Shapes Determined using ANSYS....... 64
6.2. Mode 1-Bending Mode about x-axis ............................................................. 65
6.3. Mode 2- Bending Mode about z-axis ............................................................ 65
6.4. Mode 3-Torsion Mode about x-axis .............................................................. 66
6.5. Mode 4-First Bending Mode about y-axis..................................................... 66
xv
Figure Page
6.6. Mode 5-Second Bending Mode about y-axis ................................................ 67
6.7. Graph of Natural Frequncy Squared versus Length of the Lifting Rods ...... 68
6.8. Power Spectral Density Plot of the Data Obtained in the Axial Direction of
Lifting Bar Assembly of Kingston Steam Power Plant................................. 71
6.9. Power Spectral Density Plot of the Data Obtained in Parallel with Axial
Direction Measurements from the Accelerometer Mounted on the Ground
in Kingston Steam Power Plant..................................................................... 71
6.10. Transfer Function Plot Relating the Industrial Background Vibration
Signal Data and the Axial Direction Signal Data.......................................... 72
6.11. Coherence Plot Relating the Industrial Background Vibration Signal Data
and the Axial Direction Signal Data of Kingston Power Plant ..................... 72
6.12. Power Spectral Density Plot of the Data Obtained in the Transverse
Direction of Lifting Bar Assembly of Kingston Steam Power Plant ............ 74
6.13. Power Spectral Density Plot of the Data Obtained in Parallel with
Transverse Direction Measurements from the Accelerometer Mounted on
the Ground in Kingston Steam Power Plant.................................................. 74
6.14. Transfer Function Plot Relating the Industrial Background Vibration
Signal Data and the Transverse Direction Signal Data ................................. 75
6.15. Coherence Plot Relating the Industrial Background Vibration Signal Data
and the Axial Direction Signal Data of Kingston Power Plant ..................... 75
xvi
Figure Page
6.16. Equivalent Stress Contour Plot of the Fourth Case at the End of Ramped
Loading.......................................................................................................... 79
6.17. Equivalent Stress Plot of the Fourth at the Beginning of the Steady State
Second Load Step.......................................................................................... 79
7.1. Lifting Bar Assembly Modified to Change the Flow Pattern of Steam
inside the Steam Chest .................................................................................. 83
xvii
LIST OF SYMBOLS AND ACRONYMS
Symbol Description
a1, …, a
30 Constants
{ d} Nodal displacement matrix
dt Duration of impact
d1 Damage fraction
f frequency
cf Cut off frequency
[k] Element stiffness matrix
[m] Element mass matrix
nfft zero-padded length
noverlap Amount of overlap
n1 Number of applied load cycles
pxx
Power spectral density of Signal x
t Time
u Nodal displacement in x-direction
uI Nodal displacement of I-th node in x-direction
ux, u
y, u
z Translational degrees of freedom
v Nodal displacement in y-direction
vI Nodal displacement of I-th node in y-direction
xviii
Symbol Description
w Nodal displacement in z-direction
window Size of window
wI Nodal displacement of I-th node in z-direction
x, y, z Cartesian coordinate axes
)(tx Response variable of signal x at any time, t
)t(x& Velocity response at any time, t
xI, y
I, z
I Coordinates of I-th node
)(ty Response variable of signal y at any time, t
A Amplitude
Am Amplitude factor
[B] Strain displacement matrix
Cload
Load correction factor
Csize
Size correction factor
Csurf
Surface correction factor
Ctemp
Temperature correction factor
Crelib
Reliability correction factor
Cmisc
Miscellaneous correction factor
[D] Material property matrix
E Young’s modulus
Fs Sampling rate
xix
Symbol Description
Gxx One-sided auto power spectral density
Gxy One-sided cross-power spectral density
Hxy
Transfer function
[K] Total stiffness matrix
[M] Total mass matrix
Sf Corrected fatigue strength
Sf’ Uncorrected fatigue strength
)( fX Fourier transform of signal x(t)
)( fY Fourier transform of signal x(t)
[K] Total stiffness matrix
[M] Total mass matrix
N Number of samples
N Shape function
pN Number of segments
Rxx
Autocorrelation
T Time period
)( fX Direct Fourier transform of x(t)
2xyγ Coherence
ε Strain
xε Strain in x-direction
xx
Symbol Description
ζ Critical damping ratio
θx, θ
y, θ
z Rotational degrees of freedom
xµ Average value of random signal x
ν Poisson’s ratio
ρ Mass density
aσ Alternating stress
mσ Mean stress
maxσ Maximum stress
minσ Minimum stress
utσ Ultimate tensile stress
yσ Yield strength
τ Incremental time
φ Phase angle
2ψ Variance of signal x
ω Circular natural frequency
dω Damped natural frequency
nω Undamped natural frequency
t∆ Period of signal data collection
AE Acoustic Emitter
xxi
Symbol Description
DFT Discrete Fourier Transform
FFT Fast Fourier Transform
LBV Laser Beam Vibrometer
PSD Power Spectral Density
S-N Stress at failure versus Number of cycles to failure
TVA Tennessee Valley Authority
1
CHAPTER 1
INTRODUCTION
Problem Statement
The scope of this thesis evolved from a research grant from the Tennessee Valley
Authority (TVA) to examine the cause of damage in control valve lifting bar assemblies
at the Kingston steam power plant, Kingston, Tennessee, and the Allen steam power
plant, Memphis, Tennessee. A number of control valve lifting bars in steam chests
manufactured by Westinghouse Electric Corporation have failed or have been
significantly damaged in several TVA steam plants. Steam flow-induced vibration was
suspected as the ultimate source of the damage. The lifting bar assembly is completely
embedded within the steam chest and the operating temperature is approximately 1000
ºF. Therefore, no direct method, such as the use of accelerometers, could be employed to
assess the dynamic characteristics.
The top and side views of the Westinghouse steam chest governor valves and
lifting bar assembly are shown in Figure 1.1 [1]. The parts 39 and 45 of the lifting bar
assembly are the lifting rods and the lifting bar. Two lifting bars are shown in the cross-
sectional drawing of Figure 1.1. The lifting bars are used to raise and lower the control
valve lifting bar. The dimension of the 65 in. long by 6.5 in. wide and 5 in. high valve
lifting bar assembly for the Allen steam power plant are shown in Figure 1.2 [2]. The
2
Figure 1.1. Steam Chest with Lifting Rod (Part 39) Identified in Section A-A (Drawing taken from Westinghouse I.L.1250-602) [1]
details of 43 in. long by 4.5 in. wide and 4.5 in. high valve lifting bar assembly of the
Kingston steam power plant are shown in Figure 1.3 [2].
3
Figure 1.2. Detailed Drawing of the Lifting Bar Assembly of Allen Steam Power Plant used for Finite Element Analysis (adapted from [2])
y
x
x
z
4
Figure 1.3. Detailed Drawing of the Lifting Bar Assembly of Kingston Steam Power Plant used for Finite Element Analysis (adapted from [2])
x
y
x
z
5
The lifting bar assembly is made of Refractaloy 26. Refractaloy 26 is a high heat
resistant alloy of nickel, chromium, cobalt, and iron. Refractaloy 26 has high creep
strength and fatigue endurance strength. The alloy has high ductility at elevated
temperatures. The general composition of this alloy is given in Table 1.1. A list of
physical and mechanical properties of Refractaloy 26 at the operating temperature of
1000 ºF of lifting bar assembly is given in Table 1.2. It should be noted that the
endurance limit in Table 1.2 is taken at 1200 ºF instead of 1000 ºF [3] because data at
1000 ºF were not available.
Table 1.1. Composition of Refractaloy 26 [3]
Element Percentage Nickel Cobalt
Chromium Molybdenum
Titanium Aluminum
Carbon Silicon
Manganese Iron
35.00-39.00 18.00-22.00 16.00-20.00 2.50- 3.50 2.30- 2.90 0.25 max. 0.08 max. 0.50- 1.50 0.40- 1.00 balance
Table 1.2. Physical and Mechanical Properties of Refractaloy 26 at 1000 ºF [3]
Property Value
Ultimate Tensile Strength, σut
Yield Strength (0.2 % offset), σy
Percent Elongation (2 in. gage length)
Young’s Modulus, E
Mass Density, ρ
Poisson’s Ratio, ν
Endurance Limit at 108 cycles, 1200 ºF
143 ksi
85 ksi
18
26.3 × 106 psi
7.66× 10-4 lb-s2/in
4
0.296
54 ksi
6
Thesis Outline
To determine the cause of failure, modal analysis of the lifting bar assembly was
performed using analytical and experimental methods. The technical background of
obtaining modal parameters of a generalized system using signal analysis and finite
element analysis is presented in Chapter 2. The discussion of signal analysis involves a
review of properties that characterize random signal data and the theory of converting the
time domain data into frequency domain data such as mathematical background of fast
Fourier transformation. The review of finite element analysis involves the discussion of
mathematical theory of obtaining the modal parameters of a generalized system using
global stiffness and mass matrices of finite elements. A brief theory on fatigue analysis is
presented at the end of Chapter 2. Detailed discussions of the visual inspection of wear
surfaces on control valve lifting bar of Allen steam power plant is presented in Chapter 3.
A comprehensive description of the modal analysis of the lifting bar assembly
using finite element method and the fatigue analysis of the lifting bar, both performed in
ANSYS is presented in Chapter 4. The discussion on modal analysis using the finite
element analysis involves the procedure of extracting modal parameters using the finite
element package, ANSYS. The explanation on fatigue analysis of the lifting bar assembly
using ANSYS involves the method of estimating the magnitude of impact loading on the
lifting bar assembly by assuming the damage fraction for a certain period of operation.
The experimental method to extract the natural frequencies of lifting bar assembly
for sixty-four percent output of unit 1 of the Kingston steam power plant is discussed in
7
Chapter 5. The explanation on experimental method consists of the description of
experimental setup used, method of obtaining the data, and the analysis of the data using
MATLAB.
The discussion of the results obtained from experimental, finite element
procedure, and the comparison of modal analysis results is presented in Chapter 6. The
results obtained by performing fatigue analysis in ANSYS are also discussed at the end of
Chapter 6. Finally, several conclusions and recommendations are presented in Chapter 7.
8
CHAPTER 2
TECHNICAL BACKGROUND
This thesis project combines three technical areas. First, a theoretical background
involving random data signal analysis is explained. Signal analysis of time and frequency
domain data and the transformation of time domain data into frequency domain signal
data using Fourier transformations are reviewed. Second, the basic theory and procedure
to obtain the natural frequencies and mode shapes using the finite element method is
explained. Finally, the theory and the equations governing fatigue analysis are discussed.
Signal Analysis
The following discussion of signal analysis is adapted from Bendat and Piersol
[4]. A signal is defined as any physical phenomena that occurs in common engineering
interest. A signal is usually measured in terms of a response variable varying with time.
Some response variables of interest in the thesis are displacement, velocity, and
acceleration. The signal data are generally classified into two categories: deterministic
signal and nondeterministic or random signal. A deterministic signal can be predicted
accurately at any time using an exact mathematical relationship. For example, the free
vibration of a single degree of freedom, damped system (shown in Figure 2.1) can be
mathematically modeled using [5]
),cos()( φωζω −= − tAetx dtn (2.1)
9
Figure 2.1. Mass, Spring, and Damper Single Degrees of Freedom System and its Free Body Diagram
where A is the amplitude of the signal, ωn is the undamped natural frequency, ωd is the
damped natural frequency, ζ is the critical damping ratio, and φ is the phase angle. On the
other hand, the response of the same system to random excitation cannot be predicted at
any time. The theory of random signal data is briefly explained in the following
subsections.
Random Signal Data
Random signal data cannot be defined with a mathematical equation. However,
many physical phenomena in real world are random. For such data, each set of time
domain results is unique and will not be repeated. To fully understand this kind of data, a
number of such experiments have to be conducted to produce a finite number of time
history records xi(t), i = 1, 2, 3, …, N, as shown in Figure 2.2. A collection of such time
10
Figure 2.2. Ensemble of Time History Records Defining Random Process xi(t) (adapted from [4])
history data is usually called an ensemble. An ensemble defines the random process x(t)
of the phenomenon. The characteristics of an ensemble are usually evaluated in a
statistical sense. To assess the random data properties, the ensemble average values and
the average squared values at any instance of time are used. The ensemble average is also
called the mean value, xµ , of the data. For example, at a particular time t1 (shown in
Figure 2.2), the ensemble average is calculated using
11
.)(1
lim)(1
11 ∑=∞→
=N
ii
Nx tx
Ntµ (2.2)
The ensemble average squared value is also called mean square value, 2xψ , or variance.
The mean square value at any time t1 of the data are calculated using
.)(1
lim)(1
12
12 ∑
=∞→=
N
ii
Nx tx
Ntψ (2.3)
Other properties, such as autocorrelation and higher-order average values, can be
used to characterize random data. Autocorrelation, Rxx, is calculated using
.)()(1
lim),(1
111 ∑=∞→
+=N
iii
Nxx txtx
NtR ττ (2.4)
The autocorrelation is defined as average product of the data values at time t1 and t1+τ for
the data shown in Figure 2.2. For more information on higher order average properties,
the reader should consult Bendat and Piersol [4].
If the ensemble average values of Equations 2.2 and 2.3 do not change with time,
then the random data is considered to be stationary. If these average values change with
time, then the data is considered to be nonstationary. For stationary data, the average
properties can be calculated from an individual record x(t) of length T rather than using
ensemble at time t1. Thus, the average properties for stationary random data are given as
∫∞→=
T
Tx tx
T 0
)(1
limµ ,dt (2.5)
∫∞→=
T
Tx tx
T 0
22 )(1
limψ ,dt (2.6)
12
∫ +=∞→
T
Txx txtxR
0
)()(lim)( ττ .dt (2.7)
Nonstationary data must be evaluated using Equations 2.2 through 2.4.
Frequency Domain Data
Experimental data is usually obtained in the time domain. For vibration analysis
of continuous or multi-degree of freedom systems, the natural frequencies of a system are
best determined in the frequency domain. Natural frequencies in the frequency domain
appear as peaks in the data. A brief overview of transformations of data from time
domain to frequency domain is explained in the following paragraphs.
Fourier Series. For any periodic function x(t) to be expanded as a Fourier series,
the function should obey the following conditions:
1. x(t) should have a finite number of maxima and minima within the period, T
2. x(t) should have a finite number of discontinuities within the period, T
3. x(t) should also satisfy the following equation
∫T
tx0
)( dt .∞< (2.8)
The first two conditions imply that the periodic data x(t) is integrable. The third
condition states that the integral is finite. If all three conditions are met by the periodic
function x(t), then the periodic function can be expanded in complex form as
∑∞
−∞==
m
tfjm
meAtx π2)( . (2.9)
13
For more information on Fourier series, the reader should see Thomson [6]. The
knowledge of the Fourier series as given in Equation 2.9 is used only to expand any
periodic data into a Fourier series. The following paragraphs explain the Fourier
transformation of a nonperiodic signal.
Fourier Transforms. Many phenomena occurring in the real world are
nonperiodic. For nonperiodic data, the Fourier series expansion given by Equation 2.9
has to be extended by considering what happens as the period, T, approaches infinity. The
discrete spectrum of periodic functions becomes a continuous function for nonperiodic
response. The Fourier transforms for nonperiodic random data requires the use of the
Fourier integral. The Fourier integral can be defined as the limiting case of the Fourier
series as the period approaches infinity. The Fourier integral can be calculated using
∫∞
∞−= dfefXtx fti )()( 2π (2.10)
where X(f) is called the Fourier transform of x(t), which can be calculated using
∫∞
∞−
−= ftjetxfX π2)()( ,dt ∞<<∞− f . (2.11)
Finite Fourier Transforms. For stationary random data, which is nonperiodic,
the time period of the random time history record, x(t), approaches infinity. Therefore,
the integral in Equation 2.8 is equal to infinity. Thus, the third condition necessary for
expanding x(t) as Fourier series is violated. To overcome this difficulty, the Fourier
transformation is performed for a finite interval of time, T, in which the random data are
collected. The finite Fourier transform of random signal data is
14
∫−==
Tftj
T etxTfXfX0
2)(),()( π .dt (2.12)
The finite Fourier transform of random signal data always exists for finite lengths of
measured time.
Discrete Fourier Transforms. A digital computer stores response data in
discrete form. The discrete random signal data represents a series of impulses with the
magnitude equal to the amplitude of the continuous waveform for particular time step.
When the random signal data x(t) is sampled at points ∆t apart, the record length becomes
T = N∆t, and the sampling rate is 1/∆t. The sampling rate is the number of data signals
collected per second. The sampling rate is an important factor in analyzing any random
signal data because it determines the maximum or cutoff frequency for which the finite
Fourier transform is valid. This cutoff frequency is expressed as
.2
1
tf c ∆
= (2.13)
The Fourier transformations as given by Equations 2.11 and 2.12 are valid for
continuous random data. These transformations must be altered for use on discrete
random signal data. For digitally sampled data, the discrete Fourier transform is used to
analyze the signal data in the frequency domain. The discrete Fourier transform (DFT) of
discrete random signal data can be obtained by using
∑=
−∆=∆=
N
n
N
mnj
nm extfmXX1
2)(
π, m = 1, 2, 3, …, N. (2.14)
The inverse discrete Fourier transformation of random signal data to obtain x(t) from X(f),
is
15
∑=
∆=∆=N
m
N
mnj
mn eXftnxx1
2)(
π, n = 1, 2, 3, …, N. (2.15)
To evaluate the DFT using Equation 2.14, the sum has to be performed for every
n and each sum has length N. Thus, the direct sum requires a total number of N2
operations. The fast Fourier transform (FFT) is much more computationally efficient to
evaluate the discrete Fourier transform. With the fast Fourier transform algorithm, the
sum can be performed in N log2 (N) operations. The FFT algorithm is based on the
Daniel-Loczos theorem. For more information, the reader should consult Press, et al. [7]
and Ramirez [8].
Spectral Functions. The following discussion on spectral density functions is
adapted from Meirovitch [9]. For the general case of two different measurements x(t) and
y(t), the auto-power spectral density Gxx(f) of signal x(t) can be calculated from the fast
Fourier transform X(f) using
2
)(2)( fXfGxx = . (2.16)
Similarly, the auto-power spectral density of signal y(t) can be calculated by replacing y
with x in Equation 2.16. The power spectral density function is used to determine the
natural frequencies of the systems x(t) and y(t). The cross-power spectral density Gxy(f)
can be calculated using
)()(2)( * fYfXfGxy = , (2.17)
where Y*(f) is the complex conjugate of the Fourier transform Y(f).
For random signal data, the power spectral density of the discrete time-signal
vector can be estimated using Welch’s averaged, modified periodogram method [10].
16
Welch’s averaged method involves taking the sampling function and segmenting it into S
samples. The total number of segments is then calculated as Np. These sections can also
overlap by an amount of noverlap. The FFT of each segment is calculated and multiplied
by its complex conjugate and averaged to obtain the power spectral density estimate. The
power spectral density of sequence x is calculated using
,)(1 1
2)(∑−
=pN
p
p
pxx kx
Np (2.18)
where the superscript p simply denotes the PSD of segment Np. The variance in the PSD
is reduced by the number of averages pN . The sample function is segmented using a
windowing technique.
To calculate the signal to industrial background vibration ratio, the transfer
function, Hxy(f) is calculated between the input industrial background vibration signal
data and the output response data. The transfer function can be calculated using
)(
)()(
fG
fGfH
xx
xyxy = . (2.19)
The coherence function is used to measure the statistical independence of two
different measurements x(t) and y(t). The coherence function can be calculated using the
cross-spectral density function (Equation 2.17) and the autospectral density function
(Equation 2.16). The coherence function 2xyγ can be calculated using
)()(
)()(
2
2
fGfG
fGf
yyxx
xy
xy =γ , 10 2 ≤≤ xyγ . (2.20)
17
The value of the coherence function always lies between zero and one; the zero indicates
that the signals x(t) and y(t) are unrelated and the one indicates the signals x(t) and y(t)
are related. For more information about spectrum density functions, the reader is referred
to Bendat [4].
Distributed Systems
Many physical systems cannot be modeled in a discrete manner using lumped
masses, springs, and dampers. Such systems are distributed or continuous systems and
are characterized by distributions of mass. Distributed systems have an infinite number of
degrees of freedom and therefore, have infinite number of natural frequencies. Each
natural frequency of a distributed system has a unique shape associated with its free
vibration. The shapes are referred to as modes of vibration or mode shapes.
To determine mode shapes and their accompanying natural frequencies, the set of
partial differential equations that govern the response of a distributed system must be
solved. Simple distributed structures, such as beams, have closed-form solutions for
natural frequencies and mode shapes. More complicated structures usually require
numerical solutions of the partial differential equations. The finite element method is
commonly used to analyze distributed systems.
18
Finite Element Method
A brief explanation of the finite element method used to obtain the modal
parameters of the system is presented in this section. In the finite element method, the
continuous model is divided into a finite number of discrete parts called elements. These
elements represent the spatial volume and connectivity of the actual system. Each
element will have a definite number of nodes, degrees of freedom, and shape. The
stiffness and mass of each element can be determined using the following material
properties: Young’s modulus, Poisson’s ratio, and density. After determining the element
stiffness and mass properties, the natural frequencies and mode shapes can be extracted
approximately using the assembled stiffness and mass matrices of the total system. In
addition, strains and stresses can be determined if the appropriate constraints and loads
are applied.
The formulation of stiffness matrix depends on the type of element chosen to
perform finite element analysis of the structure. The tetrahedron 10-node element, shown
in Figure 2.3, is commonly used in finite element modeling. The 10-node element was
chosen to allow the use of automatic meshing algorithm that is available in most
commercial finite element programs.
The ten-node tetrahedral element [11], shown in Figure 2.3, has three translation
degrees of freedom at each node (nodal x, y, and z directions). The unknown nodal
displacements can be represented in matrix form as
19
Figure 2.3. Ten-Node Tetrahedral Solid Element Used to Discretize the Three-Dimensional Structure [11]
{ } .
.
R
R
R
I
I
I
⋅⋅=
w
v
u
w
v
u
d (2.21)
Since there are three nodes along each edge of the tetrahedron element, the
element displacement functions u, v, and w are quadratic along each edge. The
displacement functions for the ten-node tetrahedral element are
.),,(
,),,(
,),,(
3029282
272
262
2524232221
2019182
172
162
1514131211
10982
72
62
54321
zxayzaxyazayaxazayaxaazyxu
zxayzaxyazayaxazayaxaazyxv
zxayzaxyazayaxazayaxaazyxu
+++++++++=
+++++++++=
+++++++++=
(2.22)
By substituting the known nodal coordinates (xI, y
I, z
I, … x
R, y
R, z
R) and the
unknown nodal displacements (uI, v
I, w
I, … u
R, v
R, w
R) of the element, the constants ai’s
can be evaluated in terms of nodal displacements. The shape function matrix [ ]N relates
the displacement of any point in an element with its nodal displacements. For the ten-
20
node tetrahedral element, [ ]N is a 303× matrix that is a function of x, y, z and the nodal
displacements uI, v
I, w
I, … u
R, v
R, w
R. For further details about this element, the reader is
referred to the ANSYS Theory Manual [11].
Based on the assumed displacement functions, the strain field is
{ }
+++++++
+++++++
+++++++
+++
+++
+++
=
∂∂+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂∂∂∂∂
=
=
zayaxaaxayazaa
zaxayaaxayazaa
zayaxaazaxayaa
xayazaa
zaxayaa
zayaxaa
x
w
z
uy
w
z
vx
v
y
uz
wy
vx
u
yzx
yz
xy
z
y
x
302825221094
2928262320191714
201815129863
30292724
19181613
10852
22
22
22
2
2
2
γ
γ
γ
ε
ε
ε
ε . (2.23)
The strains { }ε can be written in terms of nodal displacement matrix
{ } [ ]{ }dB=ε , (2.24)
where the matrix [B] is matrix of derivatives of [ ]N that relate the strain at any point
within an element to the nodal displacements. The matrix [B] is called the strain
displacement matrix and has the size 630× for the ten-node tetrahedral element.
To obtain stress at any point, Hooke’s law is used
{ } [ ]{ }εσ D= , (2.25)
where [D] is the material property matrix. The material matrix can be calculated if the
Young’s modulus and Poisson’s ratio of the material is known. If the material is
isotropic, then the [D] matrix for three-dimensional element is
21
[ ]
−
−
−−
−−
−+=
2
2100000
02
210000
002
21000
0001
0001
0001
)21)(1(
ν
ν
νννν
νννννν
ννE
D . (2.26)
More details on the finite element formulation for displacement, strain, and stress are
given by Logan [12].
The key quantities for modal analysis are the stiffness matrix and mass matrix.
The element stiffness matrix, k can be calculated using
[ ] [ ] [ ][ ]∫∫∫=V
T dVBDBk . (2.27)
The element stiffness matrix is a 3030× matrix for the ten-node tetrahedral element. The
element mass matrix of any element can be calculated using
[ ] [ ] [ ]∫∫∫=V
TNNm ρ dV , (2.28)
where [ ]N is the shape function matrix. Since the shape function matrix is fixed for a
particular element type, the mass matrix can be calculated if the density of the material is
known. The element mass matrix of ten-node tetrahedral element has the size 3030× .
The total stiffness matrix [K] and the total mass matrix [M] of the structure can be
obtained by assembling each of the element matrices. After obtaining the total stiffness
and the total mass matrices, the natural frequencies ω of the structure can be obtained by
solving the characteristic equation
22
02 =− MK ω . (2.29)
The mode shapes can then be determined by substituting the corresponding natural
frequency in any column of adjoint matrix of the characteristic matrix. More details of
this eigenproblem are given in Logan [12].
Fatigue Analysis
Fatigue is a damage and failure mechanism that can occur at stress levels
significantly lower than the yield strength because of the repeated application of load.
The fluctuation of load causes cracks to nucleate and grow in a machine component or
structure. A brief description of fatigue is given here. Additional information can be
obtained from texts on engineering materials such as Hertzberg [13] or machine design
such as Norton [14]. Complete treatments on fatigue are given by Suresh [15] and
Banantine, et al. [16].
There are many factors that influence the fatigue behavior. The main factors that
influence fatigue are
(1) The number of applied load cycles N,
(2) The amplitude of the applied stress aσ ,
(3) The mean stress mσ ,
(4) The presence of stress concentrations,
(5) The quality of surface finish.
23
An illustration of the stress experienced by a structure subjected to cyclic loading
is shown in Figure 2.4. In this figure, a plot of stress versus time is given. Although the
example in Figure 2.4 is a sinusoidal wave with constant amplitude and a fixed
frequency, real structures often are subjected to more complex loading. The stress
amplitude aσ is defined as one-half of the difference between the maximum stress in a
cycle maxσ and the minimum stress minσ . The mean stress mσ is the average of minσ and
maxσ .
The number of cycles N experienced by a structure before failure determines the
fatigue approach that will be used. Generally, for shorter fatigue lives (less than 1000
cycles), the strain-life approach is used. The strain-life approach must be used when the
plastic strains are comparable in magnitude to the elastic strains. For longer fatigue lives
(greater than 1000 cycles), the stress life approach is used. Often, damage fractions are
calculated to quantify the amount of damage or fatigue life that has been used. If the
Figure 2.4. Stress Varying in Sinusoidal Fashion Showing Mean and Alternating Stress Experienced by a Structure
24
number of cycles to failure for stress amplitude S1 is N
1, then any number of cycles n
1
less than N1 will not cause failure. The damage fraction d1 at this stress level is
1
11 N
nd = . (2.30)
Thus, d1 = 1 indicates that a fatigue failure has occurred.
The alternating stress aσ is an important factor in fatigue design. In the absence
of mean stress, the alternating stress cannot exceed the fatigue strength. A sketch of
alternating stress at failure Sf versus cycles to failure N
f (S-N curve) is shown in Figure
2.5. Note that the plot contains the log of cycles to failure and the log of fatigue strength.
The mean stress mσ is also very important in fatigue design. Compressive mean stresses
can improve fatigue strength. This improvement is shown schematically in Figure 2.5,
where the S-N curve for a compressive mean stress is higher than the S-N curve for a zero
mean stress. However, tensile mean stresses can degrade fatigue strength. This
degradation is shown in Figure 2.5, where the S-N curve for a tensile mean stress is lower
than the S-N curve for zero mean stress. For a fixed number of cycles to failure (see
dashed vertical line in Figure 2.5), the fatigue strength is reduced as the mean stress
increases from a compressive value to a tensile value. The mean stress is calculated using
2
minmax σσσ +=m . (2.31)
25
Figure 2.5. Alternating Stress at Failure versus Cycles to Failure
The interaction between alternating and mean stress at failure is most simply
represented by the Goodman equation
1=+ut
m
f
a
S σσσ
. (2.32)
The Goodman equation states that fatigue failure can be determined by examining
the sum of two ratios: applied alternating stress to fatigue strength and applied mean
stress to ultimate strength. The fatigue strength Sf used in the Goodman equation is the
alternating stress at failure for a specific number of cycles. The point on S-N curve where
the fatigue strength no longer changes with increasing number of cycles defines the
endurance limit Se. Some materials such as steels have endurance limits. Other materials,
such as aluminum alloys, do not have endurance limits. Instead, these materials have a
fatigue strength that continues to decrease as the number of cycles to failure increases.
The presence of stress concentrations adversely affects fatigue strength. In
addition, poor surface finish can degrade fatigue strength. In fact, many variables can
Se
26
improve or degrade fatigue strength. The effect of these variables can be quantified using
correction factors tabulated in many machine design books (see Norton [14] for
example). The following equation demonstrates how to account for the factors that
influence fatigue strength
Sf = miscrelibtempsurfsizeload CCCCCC Sf′, (2.33)
where the C’s are tabulated or approximated by curve fits for a variety of conditions and
materials. The uncorrected fatigue strength Sf′ for steels is generally calculated as half of
the ultimate strength. For other materials, Sf′ is calculated by assuming a certain fraction
of ultimate strength. There are many curve fits of correction factors available for steels.
Therefore, for other materials these correction factors are used with caution.
The load correction factor Cload
accounts for the type of loading (bending, axial,
torsion). Specifically, the load correction factor is
= torsion.& bending0.1
axial 7.0loadC (2.34)
The size correction factor Csize
accounts the difference in cross section of the part
and a test specimen subjected to bending and torsion. For example, the size correction
factor for a round section is
≤≤
≤=
− in. 103.0869.0
in 3.0 1
097.0 dd
dCsize (2.35)
Norton [14] lists size corrections for other cross sections.
27
The surface correction factor Csurf
accounts for the effect of surface finish on
fatigue strength. For example, a curve fit expression for a machined finish is given by
Norton [14] as
265.0)(7.2 −= utsurfC σ , (2.36)
where utσ is given in ksi. If the calculated surface correction factor is greater than one
using Equation 2.36, then Csurf
should be reset to one.
The temperature correction factor Ctemp
accounts for operating temperatures
greater than room temperature. The reliability correction factor Crelib
accounts for
statistical scatter in fatigue tests. For example, a reliability of 99 percent would require
Crelib
= 0.814 for steel. The correction factor Cmisc
accounts for any additional factors such
as stress concentrations. The reader should consult Norton [14] or Shigley [17] for
additional details about Ctemp
, Crelib
, and Cmisc
.
28
CHAPTER 3
VISUAL INSPECTION OF WEAR DAMAGE
Four damaged areas of the control valve assembly were found during the April 9,
1999 visit to the Allen Steam Power Plant [18]. The components had been previously
disassembled to allow close visual inspection of individual parts and subassemblies.
Visible damage was found in four areas:
(1) The underneath sides of the holes in the lifting bars,
(2) The connection of lifting rod to the lifting bar,
(3) The conical support section of the control valve, and
(4) The lateral surfaces of the lifting rod.
The first damaged area is a large arc-shaped buildup of material seen in several
holes in the lifting bars. This type of damage is shown in Figures 3.1 and 3.2. The worst
damage of this type is shown in Figure 3.1. A more typical state of damage is shown in
Figure 3.2. This buildup of material is caused by very large compressive stresses
generated when the valves impact the lifting bar. The ultimate source of this damage is
the flow-induced vibration of the valve.
The second damaged area is a metallic peen-type plastic deformation at the
interface of the lifting rod connection with the lifting bar shown in Figures 3.3 and 3.4.
The wear lines in Figure 3.3 are circular and may be attributed to a large amount of
horizontal swinging vibration. The lifting rod in Figure 3.3 is at the downstream end of
29
Figure 3.1. Large Half Moon Shaped Compression Buildup from High Velocity Momentum of the Valve
Figure 3.2. Typical Wear on the Underneath Side of the Supporting Hole Resulted from High Velocity Momentum of the Valve
30
Figure 3.3. Metallic Peen Type Deformation Observed at the Interface of the Lifting Rod Connection with the Lifting Bar
Figure 3.4. Metallic Peen Type Deformation Observed at the Interface of the Lifting Rod Connection with the Lifting Bar
31
the lifting bar. Along this part of the cylindrical cavity, the steam completes its descent to
the valve exits. The horizontal lines of wear shown in Figure 3.4 indicate that this end of
the lifting bar acted like a knife-edge support for the swinging motion of the rest of the
lifting bar. The twist-lock base connection of the lifting rod is shown in Figure 3.5. The
wear patterns on the lifting bar along the long sides of the rectangular-hole matches the
patterns shown on the lifting bars (Figures 3.3 and 3.4).
The third damaged area is the wear on the conical support section of the valve
shown in Figure 3.6. The amount of wear on this area is comparatively less than other
damaged regions, but the fact that the damage is uniform along the circumference
indicates that vibration is the likely source of the loading that caused the damage.
The fourth damaged area is the wear on the lateral surfaces of the lifting rods
shown in Figures 3.7 and 3.8. The longitudinal wear lines on the lower ends of the lifting
rods occur when there is a significant radial force that breaks through the normal steam
lubricant. The large force may be caused by dimensional changes in the lifting bar due to
thermal expansion.
The visual inspection of wear damage on April 9, 1999 at the Allen Steam Power
Plant revealed four damaged areas. TVA has previously observed damage in these four
areas during periodic maintenance of the steam chests. Three of the four damaged areas
appear to be caused by steam flow-induced vibrations. The fourth wear area on the lateral
surfaces of the lifting bars does not appear to have been damaged from vibration.
32
Figure 3.5. Wear on the Lifting Bar at the Twist Lock Base Connection of the Control Valve Lifting Rod
Figure 3.6. Control Valve in the Lifting Bar Assembly Showing Wear in All Directions on the Conical Seat in Equal Amount
33
Figure 3.7. Wear on the Lifting Rod at the End where there is a Significant Radial Force breaking through the Normal Steam Lubricant
Figure 3.8. Wear on the Lifting Rod at the End where there is a Significant Radial Force breaking Through the Normal Steam Lubricant
34
CHAPTER 4
ANALYTICAL AND NUMERICAL PROGRAM
Modal Analysis
The modal analysis of the control valve lifting bar using the finite element method
is performed either to obtain the first five modes of vibration or the modes of vibration of
that exists within 0-1250 Hz. The different frequencies obtained by solving the model
numerically will be compared with the experimental results obtained from the data
collected at TVA’s Kingston steam power plant. In this chapter, the procedure to
calculate natural frequencies of the control valve lifting bar assembly by the finite
element method is explained. In addition, the plan to infer experimental modes of
vibration from the corresponding numerical modes is discussed. Finally, the plan for
fatigue analysis of the lifting bar is explained.
The modal analysis of the control valve lifting bar using the finite element method
involves three steps. The first step is model preparation, which consists of the geometric
modeling the lifting bar, specifying material properties, selecting element types, and
meshing the model. The second step is solution which includes specifying the boundary
conditions, applying appropriate loads, choosing the relevant solver, specifying the
number of modes of vibration, and solving the model. The third step is the examination
of results which includes reviewing the results summary of the natural frequencies and
animating the different modes of vibrations experienced by the model at each power
35
output level of the Kingston plant. The modal analysis was performed using the finite
element software package ANSYS 5.5.3.
Model Preparation
Model preparation was the first step in the modal analysis of the control valve
lifting bar assembly. This step was primarily used to create a solid model of lifting bar
assembly. The lifting bar model created in this step consists of two lifting rods and a
lifting bar as shown in Figure 4.1. Since the model has irregular shapes, such as fillets,
holes, and arcs, a 10-node tetrahedral solid element was selected to discretize the model
into finite elements. The tetrahedral element (ANSYS Solid 92) employed for meshing
has three translational degrees of freedom ux, uy, and uz at each node. The three rotational
degrees of freedom θx, θy, and θz were also included to accurately simulate the boundary
conditions on the lifting rods. The finite element model, shown in Figure 4.2, consists of
3234 elements and 5682 nodes. The material properties required to perform modal
analysis (E, ρ, and ν) are given in Table 1.1.
36
Figure 4.1. Geometric Model of Control Valve Lifting Bar Assembly of TVA Kingston Power Plant
Figure 4.2. Element Plot of Control Valve Lifting Bar Assembly of TVA Kingston Power Plant
37
The free length of the lifting rods was varied in the finite element models to
account for the different power output levels. The power output level was assumed to be
directly proportional to the elevation of the lifting bar within the steam chest. Only the
free length of the lifting rods was modeled; the constrained portion of the lifting rods is
attached to the steam chest and is not free to vibrate. The lengths of the lifting rods
corresponding to power output levels of 0, 42, 46, 64, and 100 percent were calculated
using MATLAB program (given in Appendix C). These power levels coincide with the
levels where experimental data was taken. These lengths are listed in Table 4.1. Modal
analysis was performed for all the power output levels in Table 4.1.
Table 4.1. Variation of Lifting Rod Length with the Percentage Output
Percentage Output Length of Lifting Rod (in.)
0
42
46
64
100
3.760
2.761
2.618
2.2375
1.381
38
Solution
Solution was the second phase of the modal analysis of the control valve lifting
bar. In this step, boundary conditions were applied to the finite element models. The
boundary conditions were applied to simulate the lifting bar in its working orientation.
Thus, the model was constrained on the lateral surface at the top portion of the lifting rod
in all directions except the rotation in z-direction, θz. The length of lateral surface
constrained was equal to the thickness of the steam chest. This boundary condition is
shown in Figure 4.3.
Figure 4.3. Element Plot with Boundary Conditions of the Model in Working Orientation
39
To solve the model using ANSYS, either the natural frequencies occurring in the
range of 0-1250 Hz was assigned or a minimum of five modes of vibrations of the model
was assigned. The model had a large number of constraint equations; therefore, the
subspace solver was selected.
The individual models for different power outputs of the Kingston lifting bar, and
the model of disassembled Allen lifting bar were solved separately to obtain the natural
frequencies and corresponding modes of vibration.
Examination of Results
The examination of results was the third and final phase in modal analysis of the
control valve lifting bar. In this phase, the summary of results was reviewed and animated
to determine the mode shapes corresponding to the natural frequencies. Based on the
mode shapes of the lifting bar assembly, an understanding of the wear damage and the
mode shapes causing wear damage was possible.
Mode Shape Estimation
The experimental mode shapes were assumed to match the numerically
determined mode shapes for corresponding frequencies. Local peaks in the experimental
data near the numerically calculated frequencies were assumed to be the experimental
natural frequencies. The experimental axial response was compared to the numerical
40
response corresponding to the axial mode shapes. Similarly, the experimental transverse
response was compared to the numerical response corresponding to the transverse mode
shapes. If a numerically determined mode shape corresponded to a direction other than
the axial or transverse, no experimentally matching frequency was determined.
Fatigue Analysis
To determine the magnitude of the impact loading at any critical location on the
lifting bar assembly, fatigue analysis was performed using ANSYS. The first step in the
fatigue analysis was to determine the critical locations at which maximum stresses are
produced and magnitude of the stresses at these locations. For an elastic problem, the
maximum stress locations can be obtained by performing static stress analysis using
ANSYS. Since the lifting bar assembly is being subjected to plastic deformation (see
figures in Chapter 3), the maximum stress locations were obtained by performing elastic-
plastic stress analysis using ANSYS. The true stress-strain curve for Refractaloy 26 was
assumed to be a bilinear hardening curve shown in Figure 4.4. This curve was estimated
using the material properties in Table 1.2.
41
Figure 4.4. Assumed True Stress-Strain Diagram of Refractaloy 26 for Elastic-Plastic Stress Analysis in ANSYS
The second step was to generate a S-N curve for Refractaloy 26. The material
properties required for this step were the ultimate tensile strength, the yield strength, the
percent elongation, and Young’s modulus. The fatigue correction factors were
determined based on the material properties and the geometry of the structure. To include
the effect of mean stress in the S-N diagram, the maximum and minimum stresses at
critical locations were obtained from the elastic-plastic stress analysis. A MATLAB
program to generate the S-N diagram by considering the effect of mean stress is given in
Appendix D. A sample S-N diagram of Refractaloy 26 is shown in Figure 4.5. The solid
line curve represents the fatigue curve with zero mean stress and the dashed line curve
represents the fatigue curve with a 15 ksi mean stress.
42
Figure 4.5. S-N Diagram of Refractaloy 26 Generated Using MATLAB
Since the lifting bar assembly was completely embedded within the steam chest,
the load acting on the lifting bar from the vibration of the valves in normal operation is
difficult to obtain by experimental methods. To estimate these loads numerically, various
load cases were applied to the finite element model until a maximum stress state
developed at the location where plastic deformation was observed by visual inspection.
The different load cases that were applied on the lifting bar are described as follows.
In the first case, the load was applied so that vibration of the lifting bar caused the
control valve to collide with the lifting bar. This type of loading is possible because of the
large clearance in the valve mounting. This type of loading was applied based on the
assumption that the control valve is at rest throughout the operation of the unit. As the
lifting bar vibrates with its natural frequency, it experiences an impact load with the
control valve. The finite element representation of this load case is shown in Figure 4.6.
Without mean stress With mean stress
43
Figure 4.6. Pressure Loading Applied on One Side of the Interface of Control Valve and the Lifting Bar
The red-faced outlines is the pressure loading applied on the lifting bar assembly. The
constraints were applied on the lateral surface at the top portion of the lifting rod in all
directions except the rotation about the z-axis, θz. The length of the lateral surface
constrained was equal to the thickness of the steam chest. These constraints were similar
to those applied to perform the modal analysis of the lifting bar assembly.
In the second case, the load on the lifting bar assembly was applied so that the
loading simulates the rubbing action between the lifting bar and the control valve. The
finite element representation of the loads applied on the model is shown in Figure 4.7.
The black dots are the nodes where the load is applied. The arrows indicate the loading
pattern applied.
44
Figure 4.7. Force Loading Applied on One Side of the Interface of Control Valve Resting on the Lifting Bar to Simulate the Rubbing Action
The impact load acting on the lifting bar assembly due to steam pressure was not
included in the first two load cases. The material properties of the lifting bar were
assumed be the same as the lifting rods, even though the lifting bar is at a higher
temperature than the lifting rod. In the third case, the load due to the impact of the steam
on the lifting bar and the difference in the material properties was considered. The finite
element representation of loads applied on the model is shown in Figure 4.8. The black
dots are the nodes where the load is applied. The arrows indicate the loading pattern
applied. The constraints applied on this model were the same to those applied in the first
case.
45
Figure 4.8. Pressure Loading Applied on the Front Surface of the Lifting Bar to Simulate the Impact of Steam
The three cases previously discussed were analyzed statistically even though the
actual loading is dynamic. To simulate the dynamic impact of the control valve on the
lifting bar, a transient analysis was performed for the three cases described above.
The fourth loading case on the lifting bar assembly assumes the valves move in
opposite directions at any instant in time. Based on this assumption, the loading becomes
symmetric about the center of the lifting bar. Hence, only one-half of the lifting bar was
modeled. The schematic representation of the loads applied on the model is shown in
Figure 4.9. The S’s in Figure 4.9 represents the symmetric boundary conditions applied
on the model and the red-faced outline at the valve supports represent an arbitrary
pressure loading applied on the lifting bar. The constraints on the lifting rods are similar
to those applied in the first case.
46
Figure 4.9. Pressure Loading and the Symmetric Constraints Applied on the Lifting Bar Assembly
The effect of impact loading on the lifting bar was simulated by performing
transient analysis in ANSYS. The duration of impact, dt was obtained by computing the
one-fourth of the reciprocal of the natural frequency of the lifting bar. The natural
frequency corresponding to the second y-direction bending coupled with x-direction
mode was used for calculating the duration of impact. The transient analysis of the
impact loading was applied in two load steps: ramped loading (zero-maximum loading)
and steady state loading.
To determine the exact magnitude of the pressure loading on the lifting bar, the
damage fraction is calculated for the arbitrarily chosen pressure loading case using the
47
fatigue module in ANSYS. The total number of cycles experienced by the lifting bar
assembly was calculated for six months of operation. Since the lifting bar assembly had
not completely failed after six months of operation, an iterative solution was performed
by the changing the pressure until the damage fraction is achieved in the range of 0.5 to
0.7.
48
CHAPTER 5
EXPERIMENTAL PROGRAM
Preliminary Measurements of Disassembled Lifting Bar
The disassembled left and right side lifting bars with the lifting rods in position
are shown in Figure 5.1. The four valves and the lifting rods were placed in the lifting bar
and the valves were numbered 1 through 4 starting from the nearest valve of the right side
lifting bar assembly (see Figure 5.1). Preliminary measurements were made to detect the
transverse natural frequencies by manually shaking the different valves within their
cavities.
Figure 5.1. Experimental Setup to obtain the Vibration Response of Lifting Bar Assembly when Lifting Bar is Out Side of the Steam Chest
49
Two different accelerometers were used and the largest frequencies were
measured using a Tektronix oscilloscope. First, the accelerometer was mounted on the
lifting rod near the output side (the end nearer to the valve 1) and the readings were taken
by sequentially impacting each of the four valves. Next, the accelerometer was mounted
on the lifting rod near the governor side (the end nearer to the valve 4) and the readings
were taken by manually shaking only valve 4. The results obtained from the preliminary
measurements were used to choose the appropriate sampling rate for later experiments.
Experimental Determination of Natural Frequencies
The extraction of the modal parameters of the control valve lifting rod using
experimental methods involves several issues. These issues are the selection of
appropriate transducers, the selection of proper equipment, the selection of proper
location on the lifting bar at which the data to be collected, proper mounting of
accelerometers to obtain high quality data, and the analysis of the data. These issues are
discussed in the following sections.
Selection of Transducers
Several types of transducers were considered including an accelerometer, an
acoustic emitter detector (AE detector), and a laser beam vibrometer. The usage and
selection criteria for each of these devices are explained in the following subsections.
50
Accelerometer
To obtain acoustic emission response of lifting bar assembly and industrial
background vibration response of ground, an accelerometer is used as a transducer that
converts acceleration response of the mechanical equipment or ground to electrical
voltage signal response. The selection of accelerometer depends on the several
requirements, such as the sensitivity, frequency bandwidth, dynamic range,
environmental integrity, resolution, weight, and cost. For more information, the reader
should consult Robinson and Rybak [19]. The evaluation of the above specifications for
selecting accelerometers is explained in the following paragraphs.
Sensitivity. Sensitivity of an accelerometer can be defined as the magnitude of
voltage response that can be obtained per unit gravity acceleration of the machine
component. The required sensitivity can be established by considering the response level
of the lifting bar assembly and the ground. Since the vibration response of the lifting bar
assembly and the ground surrounding the steam chest were considerably high, an
accelerometer with a sensitivity of 100 mV/g was selected to carry out the proposed task.
Frequency bandwidth. Frequency bandwidth of an accelerometer is defined
as the range of frequency of the response signal that a transducer can successfully capture
the response signal and convert it into an electrical signal. The required frequency
bandwidth of an accelerometer depends on the estimated range of vibration of the
machine component that needs to be captured. Since the vibration range of the lifting bar
51
assembly was an unknown quantity, an accelerometer with a wide range of 50 to 16000
Hz was selected to perform the required task.
Environmental integrity. The environmental integrity factor states the ability
of an accelerometer to withstand hostile environment applications. The hostile
environment of concern for monitoring the vibration response of the lifting bar assembly
is temperature. Since the temperature inside of the steam chest is very high, the
accelerometer is mounted on a magnet with a thermal insulator separating the
accelerometer and magnet. The magnetic thermally isolated mounting should not affect
the accelerometer performance.
The properties of the selected accelerometer are given in Table 5.1. Since it is
very difficult to obtain accelerometers that operate at 1000 oF, the accelerometers are
employed to obtain only industrial background vibration signal data by mounting the
accelerometer on the ground near steam chest.
Table 5.1. General Properties of the Accelerometer
PROPERTY VALUE
PCB MODEL
SENSITIVITY
FREQUENCY BANDWIDTH
TEMPERATURE RANGE
RESOLUTION
370 A02
100 mV/g
50 to 16000 Hz
-40 to 185 oF
70 µg rms.
52
Acoustic Emitter Detector
An AE detector is employed to determine both vibration response and acoustic
emissions response of sliding contact of one metal against another. It is primarily
intended to determine onset and location of cracking in materials subjected to various
loading. If any cracking or sliding contact exists, the associated energy can be found by
analyzing the electrical signals generated by AE detector. The electrical signals generated
were converted into a data file using a Tektronix oscilloscope. The AE detector required
a physical contact with the lifting bar assembly to obtain the vibration response data.
The summarized properties of the acoustic emitter detector selected to obtain
vibration response data is given in Table 5.2. The temperature of the steam chest is
approximately 1000 oF; therefore, the thermal isolation material used to mount the AE
detector was melted after only a few readings. Therefore, the data obtained by this device
were considered erroneous and so no further analysis of the AE detector data were
performed.
Table 5.2. General Properties of the Acoustic Emitter Detector
PROPERTY VALUE
DIGITAL WAVE MODEL
PRE AMPLIFIER
FREQUENCY RANGE
B105
PA2040G/A
50 kHz to 1.5MHz
53
Laser Beam Vibrometer
The laser beam vibrometer (LBV) was used to measure the vibration response of
lifting bar assembly in the incident beam direction. The LBV measured the relative
change in the distance traveled by the incident beam and reflected beam due to vibration
of lifting rod assembly in the incident beam direction. To obtain the vibration response in
both the axial and transverse directions, the measurements were taken by pointing the
laser beam in both the axial and transverse direction. The axial and transverse directions
are shown in Figure 5.2. Data files of these measurements were obtained by connecting
the laser beam vibrometer to the Tektronix oscilloscope. Unlike the acoustic emitter
detector, the LBV does not require direct contact with the lifting bar assembly. This
characteristic is the principal advantage of using the LBV in a high temperature
environment. The summarized properties of laser beam vibrometer selected to obtain
vibration response data are given in Table 5.3.
Figure 5.2. Schematic Representation of Measurement Directions for Laser Beam Vibrometer
54
Table 5.3. General Properties of Laser Beam Vibrometer
PROPERTY VALUE
OMETRON MODEL
SENSITIVITY
VIBRATION FREQUENCY
WEIGHT
VH300
3.33 mV per mm/s
DC to 25 kHz
3.7 kg (8.2 lb.)
Selection of Proper Location
The only possible location for measuring the acceleration of the valve lifting rod
was at the end of the steam sleeve bearing where the lifting rod enters the ambient air. A
simultaneous accelerometer measurement was made on the outside of the steam sleeve
for subtracting out ambient machine vibrations. The location of the measurements was
too hot for the accelerometer mounted on a magnetic attachment and for the acoustic
emission transducers mounted on a magnetic base. After the first few measurements
made near the steam sleeve bearing, these transducers were moved to cooler locations
marked with a circle in Figure 5.3.
The primary goal of the measurements was to detect dominant resonant
frequencies that were in the machine components. The handicap for all of the
measurement tools was that there was a predominant random vibration caused by the
steam flow in the governor steam chest. Random signal processing methods were
employed to bring out the deterministic steady vibration sources. The final method of
data acquisition was to collect 2,500 data points on each of the four channels for a period
55
Figure 5.3. Experimental Setup made to obtain the Vibration Response of Lifting Bar Assembly
of 1 second (i.e., each point is collected at an interval of 0.4 milliseconds). To reduce the
error resulting from the finite collection of data signals, the data was collected 10 to 15
times at the same location before moving to a new test location.
Experimental Setup
Based on the difficulty in using the AE detector, further use of the detector was
discontinued in favor of the laser beam vibrometer. The laser beam vibrometer was
56
mounted on a rigid tripod support stand with adjustable height as shown in Figure 5.3.
The vibrometer was suspended on three springs to isolate the LBV from the floor
vibrations. The approximate natural frequency of this isolation is 3 Hz. The red dot in
Figure 5.3 is the location at which the vibration response data was collected. Another
smaller tripod with a mirror was mounted inside the governor house to reflect the laser
beam around obstacles and to get measurements at right angles to those measurements
made from the door opening as shown in Figure 5.4. The mirror was also isolated from
the foundation vibrations using springs with a natural frequency of approximately 5 Hz.
Figure 5.4. Experimental Set Up to Measure the Response of Lifting Bar Assembly at Right Angles to the Door Opening Direction
57
The laser beam vibrometer was directed in the transverse direction as shown in
Figure 5.3. The data obtained from this setup were the transverse direction vibration data.
The vibrometer was connected to a Tektronix oscilloscope (not shown in Figure 5.3) to
generate a data file of the electrical signal output of laser vibrometer. The setup shown in
Figure 5.4 collects the vibration response data in the axial direction of lifting bar
assembly. The directions are also shown in the schematic drawing of Figure 5.2.
Data Signal Generation
The data signal generation involved the production of signal data in the axial and
transverse directions of the lifting bar assembly and the industrial background vibration
signal data. A laser beam vibrometer was used to produce the vibration response data
while an accelerometer was used to obtain industrial background vibration signal data.
The laser beam vibrometer and the accelerometer were further connected to a Tektronix
oscilloscope to obtain the data files. All the measurements were recorded in the time
domain. To transform the time domain data into frequency domain data, the Power
Spectral Density algorithm of MATLAB was used. The usage and essential arguments
required for this program are explained at the end of this chapter.
The time domain data files were obtained at a sampling rate of 2500 Hz at fifteen
different times to obtain fifteen different data sets for each of the axial and transverse
directions of the lifting bar assembly. Using Equation 2.13, the maximum natural
frequency that can be successfully estimated was 1250 Hz.
58
Since the period of the signal data determines the lowest frequency that can be
resolved from the FFT spectrum, the length of the time record is also an important factor
while collecting the data. If the period of the input signal is longer than the time record,
then there will be no way the period can be determined. Since frequency is defined as the
reciprocal of period, the lowest frequency that can be determined will depend on the time
record length. Therefore, the lowest line of the frequency spectrum occurs at frequency
equal to the reciprocal of the time record length.
The experimental data were stored in ASCII text files consisting of five columns
of information. The first column is the integer number that describes the number of
samples collected. The second column represents the time data that describes the time at
which the data is collected starting from zero. By subtracting any two consecutive
elements of time column, the time, ∆t, period at which each data was collected can be
determined. The third column of the data file is the signal data that represents the
vibration response of lifting bar acquired from the laser beam vibrometer. The fourth
column is the same as the second column (time data). The fifth column of the data file is
the industrial background vibration data, which represents the vibration response of the
ground acquired from the accelerometer. A sample data file is given in Appendix B.
Analysis of Experimental Data
The vibration response signal data obtained from the LBV was converted to a data
file using a Tektronix oscilloscope. A MATLAB program, given in Appendix B, was
59
used to convert the time domain data into signal plots in frequency domain. The results
are shown in a plot of amplitude versus frequency (PSD plot). The step by step procedure
of input of data file and analyzing the data of the data file to obtain PSD plots using
MATLAB program is explained as follows.
The fifteen different data files obtained in the axial direction for the sampling rate
of 2500 Hz contains signal data variation with time. These time domain data were
converted into signal vector by making it a matrix containing five columns composed of
two time columns, one signal data column, one industrial background vibration signal
data column, and a serial number column in each data file. The signal vector matrix was
then used as an input to the MATLAB program.
The MATLAB program evaluates the fifteen different data files and converts
them into a single data file containing three columns. After conversion, the first column
data represents the time data, the second column data represents the vibration response of
lifting bar, and the third column data represent industrial background vibration signal
data. The first column of individual data signal files is the serial number, which was used
to determine the number of data read by the MATLAB program.
The format in MATLAB for obtaining power spectral density [20] of the
sequence x is
),,,,(],[ noverlapwindowFsnfftxpsdfpxx = , (5.1)
where xxp is power spectrum of the sequence x, f is the frequency corresponding to the
power spectrum, x is a discrete time signal vector, nfft is the zero-padded length, Fs is
60
sampling rate, window is the length of segmented sequence x, and noverlap is the amount
of overlapping used while averaging.
The discrete time-signal vector x is divided into overlapping sections by an
amount of noverlap and windowed by a window parameter, then zero-padded to length
nfft. The FFT of each section is calculated and multiplied by its complex conjugate and
averaged to obtain power spectral density estimate, pxx
. The function PSD in MATLAB
returns both pxx
and the vector of frequencies. The power spectral density is estimated for
the vibration response of lifting bar assembly and industrial background vibration using
the Welch’s averaged modified periodogram method as described Chapter 2 (see
Equation 2.18).
After calculating the PSD, several plots were produced to obtain the natural
frequencies of lifting bar assembly. Plots of power spectral density estimate of the
response data and industrial background vibration data were produced to obtain natural
frequencies of lifting bar assembly. The natural frequencies were determined by visually
choosing the frequency corresponding to peaks observed in the PSD magnitude plots that
are close to the numerically determined frequencies. The transfer function was plotted to
determine the signal to industrial background vibration ratio in the vibration response
data. The coherence function was plotted to determine if any relation between the
vibration response data and industrial background vibration signal data exists. A
discussion of the results obtained from the experimental data is given in Chapter 6.
61
CHAPTER 6
RESULTS AND DISCUSSION
This chapter contains the results of the numerical and experimental work
performed in this thesis. The numerical results for the lifting bar in its working
orientation are presented first. After the numerical results, the experimental results are
presented. First, the preliminary measurements taken at the Allen steam plant are
discussed. Then, the experiment measurements taken at the Kingston steam plant are
discussed. Next, the numerical and experimental frequencies of the Kingston lifting bar
assembly are compared. Experimental mode shapes are assigned based on corresponding
numerical mode shapes. Finally, the results of the fatigue analysis are presented.
Numerical Results
The Kingston lifting bar assembly was analyzed in its working orientation. The
results are tabulated in Tables 6.1 through 6.5. The first column in these tables is the
mode number. The second column is the natural frequency corresponding to a mode. The
third column is a description of the mode found by animating the results in ANSYS. The
observed modes of vibration are schematically shown in Figure 6.1. This figure shows
the front, top, and right side views of the lifting bar assembly and the two extreme
positions corresponding to each mode shape. The solid red line represents one extreme of
the mode shape and the dashed red line represents the other extreme of the mode shape.
62
Table 6.1. Natural Frequencies Obtained from Modal Analysis of Control Valve Lifting Bar for 0 Percent Output of Kingston Steam Power Plant
Mode Number Natural Frequency (Hz) Observed Mode of Vibration
1
2
3
4
5
124
389
460
584
604
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
Table 6.2. Natural Frequencies Obtained from Modal Analysis of Control Valve Lifting Bar for 42 Percent Output of Kingston Steam Power Plant
Mode Number Natural Frequency (Hz) Observed Mode of Vibration
1
2
3
4
5
184
530
589
666
671
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
Table 6.3. Natural Frequencies Obtained from Modal Analysis of Control Valve Lifting Bar for 48 Percent Output of Kingston Steam Power Plant
Mode Number Natural Frequency (Hz) Observed Mode of Vibration
1
2
3
4
5
194
547
590
676
698
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
63
Table 6.4. Natural Frequencies Obtained from Modal Analysis of Control Valve Lifting Bar for 64 Percent Output of Steam Kingston Power Plant
Mode Number Natural Frequency (Hz) Observed Mode of Vibration
1
2
3
4
5
231
594
604
707
782
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
Table 6.5. Natural Frequencies Obtained from Modal Analysis of Control Valve Lifting Bar for 100 Percent Output of Kingston Steam Power Plant
Mode Number Natural Frequency (Hz) Observed Mode of Vibration
1
2
3
4
5
459
673
831
852
989
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
The first mode in Figure 6.1 is the bending of the lifting rods about x-axis. This
mode shape resembles a cantilevered beam with a concentrated mass on the free end. The
second mode is the bending of the lifting bar about the z-axis. The third mode is the
twisting of the lifting bar about the x-axis. The fourth mode is the bending of the lifting
bar about the y-axis. The fifth mode is a second bending mode of the lifting bar about the
y-axis. The numerical mode shapes corresponding to the 64 percent output are plotted in
Figures 6.2 to 6.6. These figures show the reference coordinate system used and list the
natural frequencies. The figures also show the undeformed and deformed shape of the
64
lifting bar assembly. The deformed shapes in Figures 6.2 to 6.6 are the extreme positions
of the mode shapes.
y
x
z
x
z
y
y
x
z
x
z
y
Mode 1 Bending about x-axis in the y-z Mode 4 First bending mode about y-axis plane in the x-z plane
y
x
z
x
z
y
y
x
z
x
z
y Mode 2 Bending about z-axis in the x-y Mode 5 Second bending mode about y- plane axis in the x-z plane
y
x
z
x
z
y Mode 3 Torsion about x-axis
Figure 6.1. Schematic Representation of Mode Shapes Determined using ANSYS
65
Figure 6.2. Mode 1-Bending Mode about x-axis
Figure 6.3. Mode 2- Bending Mode about z-axis
66
Figure 6.4. Mode 3-Torsion Mode about x-axis
Figure 6.5. Mode 4-First Bending Mode about y-axis
67
Figure 6.6. Mode 5-Second Bending Mode about y-axis
Frequency squared versus modeled length of the lifting rod curves are given in
Figure 6.7. These curves are approximately hyperbolic in shape. The hyperbolic shapes
are similar to the shape of frequency squared versus length plots of a pendulum or a
cantilevered beam. For a simple pendulum of length, L, the frequency is related to 1/L.
For a cantilevered beam of length, L, the frequency is related to 1/L3. The hyperbolic
shape of the curves in Figure 6.7 was consistent with the anticipated results.
68
Figure 6.7. Graph of Natural Frequncy Squared versus Length of the Lifting Rods
Experimental Results
Preliminary Measurement Results at the Allen Steam Plant
The preliminary measurements of lifting bar assembly are shown in Tables 6.6
and 6.7. The first column of these tables is the number of the valve used for impact
excitation of the lifting bar. The second column of these tables is the number of the
accelerometer used to obtain vibration response. Two sets of data were collected for each
accelerometer to reduce the experimental error while calculating the natural frequencies
of disassembled lifting bar. The lifting bar assembly was excited by manually shaking
different valves individually. The tables show the natural frequencies in the last four
columns.
69
Table 6.6. Impact Transverse Natural Frequencies (Precision: +/-5 Hz) when Accelerometers are Mounted on the Output Side of Lifting Rod
Valve Impact Location Natural Frequency (Hz)
Valve Number
Accelerometer First Second Third Fourth
900 1490 1620 First
890
550 900 1830
1
Second
950 1280 1880
890 1480 First
915
885
2
Second
340 860 1540 1860
355 870 1545 1860 First
890 1500
3
Second 350 863 1900
First 888 1492
348 863 2450
4
Second
344 864
Table 6.7. Impact Transverse Natural Frequencies (Precision: +/-5hz) when Accelerometers are Mounted on the Governor Side of Lifting Rod
Valve Impact Location Natural frequency (Hz)
Valve Number
Accelerometer First Second Third
First 205 905 1615 4
Second 208 820 1810
70
The first three transverse natural frequencies were less than 1800 Hz. Since the
lifting bar in operation is probably less constrained than the disassembled lifting bar
resting on the floor, the natural frequencies in the working orientation should be less than
the these values. Therefore, a sampling rate of 2500 Hz was chosen to collect response
data in working orientation.
Measurements at the Kingston Steam Plant
Axial response. The PSD of the vibration response in the axial direction
versus the frequency is given in Figure 6.8. The power spectral density of the
accelerometer response of industrial background vibration signal acquired simultaneously
with the axial direction data versus the frequency is given in Figure 6.9. The magnitude
of the transfer function between these two signals versus frequency is depicted in Figure
6.10. The magnitude of the coherence function of the industrial background vibration
signal and axial response signal versus frequency is shown in Figure 6.11. All results
shown are for the 64 percent output level.
The axial response PSD plot in Figure 6.8 reveals a few large peaks in the range
of 0 to 100 Hz. Another large peak occurs near 500 Hz. Three peaks occur near 700 Hz
and two large peaks occur in the range of 1000-1100 Hz. These experimental frequencies
can only be compared to the fourth and the fifth numerical natural frequencies because
the corresponding mode shapes of these two numerical frequencies have the lifting rods
71
bending in the x-z plane. The peaks that occurred nearer to these numerical frequencies
(717 Hz and 755 Hz) were considered for comparison.
0 200 400 600 800 1000 1200 140010-2
10-1Pyy - Y PSD Plot of KING UNIT1 AXIAL 64 PERCENT SAMPLE 1-15
Frequency
PS
DO
met
ron,
velo
ccity
sq /
Hz
Figure 6.8. Power Spectral Density Plot of the Data Obtained in the Axial Direction of Lifting Bar Assembly of Kingston Steam Power Plant
0 200 400 600 800 1000 1200 140010-3
10-2
10-1
100
101
102Pxx - Power Spectral Density Plot of Noise Signal Data
Frequency
PS
D P
CBA
ccel
, acc
eler
atio
nsq
/ H
z
Figure 6.9. Power Spectral Density Plot of the Data Obtained in Parallel with Axial Direction Measurements from the Accelerometer Mounted on the Ground in Kingston Steam Power Plant
72
0 200 400 600 800 1000 1200 140010-3
10-2
10-1
100Txy - Transfer function magnitude
Mag
nitu
de
Frequency
Figure 6.10. Transfer Function Plot Relating the Industrial Background Vibration Signal Data and the Axial Direction Signal Data
0 200 400 600 800 1000 1200 14000
0.1
0.2
0.3
0.4
0.5
0.6
0.7Cxy - Coherence
Mag
nitu
de
Frequency
Figure 6.11. Coherence Plot Relating the Industrial Background Vibration Signal Data and the Axial Direction Signal Data of Kingston Power Plant
73
The coherence plot in Figure 6.11 revealed a coherence of approximately 0.6 for a
single peak in the 0-100 Hz frequency range. The rest of the data has a coherence in the
0-0.3 range. Based on coherence values, the lower frequencies (peaks in the 0-100 Hz
range in Figure 6.8) are assumed to be associated with the industrial background
vibration. Thus, the lower frequency peaks in Figure 6.8 are excluded as possible natural
frequencies. The peaks in Figure 6.8 near 500 Hz and in the range of 1000-1100 Hz do
not have numerical counterparts. These experimental frequencies do not have numerical
counterparts. These frequencies should be investigated further.
Transverse response. The PSD of response in transverse direction versus
the frequency is illustrated in Figure 6.12. The PSD of the accelerometer response of
industrial background vibration signal acquired simultaneously with the transverse
direction data versus the frequency is illustrated in Figure 6.13. The magnitude of transfer
function between these two signals versus frequency is depicted in Figure 6.14. The
magnitude of coherence function of the industrial background vibration signal and
transverse response signal versus frequency is shown in Figure 6.15. All the results are
shown for 64 percent output.
74
0 200 400 600 800 1000 1200 140010-2
10-1
100Pyy - Y PSD Plot of KING UNIT1 TRANS 64 PERCENT SAMPLE 1-15
Frequency
PS
DO
met
ron,
vel
ocity
sq /H
z
Figure 6.12. Power Spectral Density Plot of the Data Obtained in the Transverse Direction of Lifting Bar Assembly of Kingston Steam Power Plant
0 200 400 600 800 1000 1200 140010 -4
10 -3
10 -2
10 -1
100
101
102Pxx - Power Spectral Density Plot of Noise Signal Data
Frequency
PS
D P
CB
Acc
el, a
ccel
erat
ion
sq /
Hz
(0.1v/g)
Figure 6.13. Power Spectral Density Plot of the Data Obtained in Parallel with Transverse Direction Measurements from the Accelerometer Mounted on the Ground in Kingston Steam Power Plant
75
0 200 400 600 800 1000 1200 140010
-2
10-1
100
101
Txy - Transfer function magnitude
Frequency
Mag
nitu
de
Figure 6.14. Transfer Function Plot Relating the Industrial Background Vibration Signal Data and the Transverse Direction Signal Data
0 200 400 600 800 1000 1200 14000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Cxy - Coherence of Transverse and Noise Signal Data
Mag
nitu
de
Frequency
Figure 6.15. Coherence Plot Relating the Industrial Background Vibration Signal Data and the Axial Direction Signal Data of Kingston Power Plant
76
The transverse response PSD plot shows the various peaks occurring at different
frequencies. Specifically, a few large peaks occur at less than 200 Hz. Beyond 200 Hz,
the peaks occur at approximately regular intervals. It is difficult to determine the natural
frequencies from the PSD plot alone because of the similar magnitudes of all the peaks
greater than 200 Hz. These experimental frequencies can only be compared to the first
three numerical natural frequencies because the corresponding mode shapes of these
three numerical frequencies have the lifting rods bending in the y-z plane. The peaks that
occurred nearer to these numerical frequencies (268 Hz, 565 Hz, and 681 Hz) were
considered for comparison.
The coherence plot in Figure 6.15 revealed a coherence of approximately 0.5 for
two peaks in the 0-200 Hz frequency range. The rest of the data has a coherence in the 0-
0.22 range. Based on coherence values, the lower frequencies (peaks in the 0-200 Hz
range in Figure 6.12) are assumed to be associated with the industrial background
vibration. Thus, the lower frequency peaks in Figure 6.12 are excluded as possible natural
frequencies.
Comparison of Natural Frequencies
Since experimental data were obtained only in the axial and transverse directions
of the lifting bar assembly, a comparison to the FEA results was made only in those
directions. The comparison of experimental results obtained from PSD plots of the
77
Kingston steam power plant and the numerical values calculated using ANSYS 5.5.3 is
shown in Table 6.8.
The bending about the x-axis occurred at a frequency of 232 Hz in the ANSYS
model, while the experimental frequency nearest this numerical frequency in PSD plot
occurred at 268 Hz. The bending about z-axis axis occurred at a frequency of 594 Hz in
the ANSYS model. The closest experimental frequency was 565 Hz. The torsion mode
about the x-axis occurred at a frequency of 604 Hz in the ANSYS model, while the
closest corresponding frequency determined experimentally was 681 Hz. The first
bending about y-axis occurred at a frequency of 707 Hz. The closest experimental
frequency was 717 Hz. The second bending about y-axis occurred at a frequency of 782
Hz in the ANSYS model. The closest experimental frequency was 755 Hz.
Table 6.8. Comparison of Natural Frequencies Obtained from Numerical and Experimental Results for Unit 1 of Kingston Steam Power Plant for 64% Output
Frequency (Hz) S.No
Numerical Value Experimental Value
Corresponding Mode of
Vibration
1
2
3
4
5
232
594
604
707
782
268
565
681
717
755
Bending about x-axis
Bending about z-axis
Torsion about x-axis
First bending about y-axis
Second bending about y-axis
78
Fatigue Analysis Results
The actual wear patterns of the lifting rods are in the axial direction of the lifting
rods. The five modes determined in the numerical analysis can all contribute to this
damage because the lifting rods being bent by all five modes. The actual wear patterns at
the valve support holes appear to be caused primarily by the second mode (bending of the
lifting bar about the z-axis). Magnitude of the load causing this wear damage was
estimated by performing fatigue analysis.
The results obtained for the first three load cases as described in the fatigue
analysis section of Chapter 4 did not show the maximum stresses at the intended location.
Thus, the assumptions made for loading the structure in these cases may not actually
resemble the true loading pattern. Therefore, these results were of no value and are not
presented in this thesis.
The results obtained for the fourth loading case, as described in fatigue analysis
section of Chapter 4, are illustrated in Figures 6.16 and 6.17. The equivalent stress
contour plot of the lifting bar assembly at the end of the ramped loading is shown in
Figure 6.16. The equivalent stress contour plot of the lifting bar assembly at the
beginning of the steady state loading is shown in Figure 6.17. The ANSYS results for this
case revealed that the maximum stresses are on the lifting rod and the valve supporting
holes. Therefore, the assumed loading pattern and geometry of the fourth loading case
were consistent with the actual impact loading of the lifting bar assembly.
79
Figure 6.16. Equivalent Stress Contour Plot of the Fourth Case at the End of Ramped Loading
Figure 6.17. Equivalent Stress Plot of the Fourth at the Beginning of the Steady State Second Load Step
80
The fatigue analysis results in Figures 6.16 and 6.17 were obtained using an
arbitrary pressure loading of 3000 psi. The results show a maximum equivalent stress of
20.6 ksi at valve supporting holes (see Figure 6.17). Using the S-N diagram in Figure 4.5,
the maximum number of repeated stress cycles the lifting bar can withstand before failure
was calculated to be 10107.1 × cycles. Using 15 lb-mass for the control valve and area of
valve supporting hole on which pressure loading was applied on the numerical model, the
acceleration of the valve was calculated to be 8 g.
It was postulated that mode 2 (bending about z-axis) was the cause of the failure
of the lifting bar at the valve supporting holes. Therefore, the natural frequency
corresponding to the second mode of vibration was used to calculate the number of stress
cycles. The number of stress cycles experienced by the lifting bar assembly for six
months of operation was calculated to be 9103.9 × based on the second natural frequency
of approximately 600 Hz. The damage fraction was then calculated using Equation 2.30.
The damage fraction for the arbitrary 3000 psi pressure loading was 0.56. This damage
was in the assumed range of 0.5-0.7 previously proposed in Chapter 4 as the observed
damage level. Therefore, the magnitude of pressure loading on the lifting bar due to the
impact of the control valves was estimated as 3000 psi.
81
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
The damage inspection revealed that the main region of dynamic wear and
damage is at the interface of the valve body and the cavity in the lifting bar supporting
the valve. The experimental vibration study shows that there is significant vibration
energy in the steam chest generator. It is hypothesized that the sources of the vibration
energy are the bending mode natural frequencies of the lifting bar assembly and the
lifting rods.
The bending modes of the lifting bar assembly are the major contributors to the
measured vibration. The valves appear to be vibrating like pendulums about their conical
support seats. When the valve hits a stop at the edge of the hole, the vibration begins. The
large clearance of the valve support for the present Westinghouse governor design and
the freedom to swing like a pendulum was the identified source of the wear and damage.
The analysis of experimental data showed several peaks in the PSD plots.
However, only experimental peaks near the numerical values were selected to compare
with the numerical values. The mode shapes of these frequencies were assumed to be the
same as the numerical mode shapes. In addition, the experimental results included several
frequencies that were not found in the numerical analysis.
The following three recommendations are given. The first recommendation is to
acquire more test data of the lifting bar assembly in operation. The second
recommendation is to explore the possibility of changing the damping characteristics of
82
the existing design. The third recommendation is to modify shape of the upstream end of
the lifting bar to change the flow characteristics in the steam chest. The detailed
explanation of these recommendations is explained as follows.
The difficulties in choosing the experimental natural frequencies from the PSD
plot revealed the need for more data sets to better understand the random response of the
lifting bar assembly in operation. Thus, collecting more data sets in operation and
analyzing these data to determine the experimental natural frequencies from PSD plot
alone is highly recommended.
The vibration of the lifting bar assembly may be reduced if the energy of the
resonant vibrations is absorbed by providing damping in the structure. Changes in
geometry or material may improve the damping characteristics.
The steam flow pattern in the steam chest could be modified by changing the
existing design. An example is given in Figure 7.1. The results obtained by performing
modal analysis of a modified lifting bar assembly using the finite element method are
tabulated in Table 7.1. The total mass of the system and size of the redesigned model was
not significantly changed. Therefore, the numerical natural frequencies of the modified
lifting bar assembly were almost the same as the original lifting bar assembly. However,
the flow pattern inside the steam chest for the redesigned model may change the
excitation frequencies of the steam flow acting on the modified lifting bar assembly. A
prototype of a modified lifting bar assembly should be further analyzed.
83
Figure 7.1. Lifting Bar Assembly Modified to Change the Flow Pattern of Steam inside the Steam Chest
Table 7.1. Natural Frequencies of the Modified Lifting Bar Assembly
Natural Frequency (Hz) Mode Number
Modified Original
1
2
3
4
5
245
553
564
682
754
231
594
604
707
782
84
REFERENCES
85
1. Westinghouse I.L. 1250-602, Steam Chest Assembly Drawing, nodate.
2. Tennessee Valley Authority, Pro/Engineer drawings of Westinghouse lifting bar assembly, no date.
3. Alloy Digest, “Refractaloy 26,” Data On World Wide Metals And Alloys, ASM International, 1996.
4. Bendat, J.S., and Piersol, A.G., Engineering Applications of Correlation and Spectral Analysis, John Wiley & Sons, Inc., 2nd Ed., 1983.
5. Derakshan, O.S., Some Studies on Parameter Identification of Linear and Nonlinear Vibration Systems, Ph.D. Dissertation, Tennessee Technological University, Cookeville, Tennessee, 1994.
6. Thomson, W. T., Theory of Vibration With Applications, Prentice Hall, Englewood Cliffs, 1988.
7. Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., Numerical Recipes, 2nd edition, Cambridge University Press, 1992.
8. Nyborg, Dan, The Fast Fourier Transform and its use in Spectral Analysis of Digital Audio, Concordia University, Online. Internet. April 22, 1998. Available: http://bohr.concordia.ca/~grob/298/fft2_paper.html.
9. Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, Inc. 1975
10. Welch, P.D., “The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodograms,” IEEE Transactions of Audio Electroacoustics, Vol. 15, pp. 70-73.
11. SAS Inc., ANSYS Theory Reference, Release 5.6 Eleventh Edition, 1999.
12. Logan, D. L., A First Course in the Finite Element Method Using Algor, PWS Publishing Company, Boston, MA, Copyright 1997.
13. Hertzberg, R. W., Deformation and Fracture Mechanics of Engineering Materials, John Wiely & Sons, Inc. 1996.
86
14. Norton, R. L., Machine Design An Integrated Approach, Prentice-Hall International Inc. 1996.
15. Suresh, S., Fatigue of Materials, Cambridge University Press, 1998.
16. Banantine, J. A., Comer J. J., and Handrock, J. L., Fundamentals of Metal Fatigue Analysis, Prentice Hall, 1990.
17. Shigley, E. J., and Mischke, R. C., Mechanical Engineering Design, McGraw-Hill, Inc., 1998.
18. Houghton, J. R., Cunningham, G. T., and Wilson, C. D., Vavilala, R., On –Line Vibrations and Acoustic Emissions Monitoring of Tennessee Valley Authority Wesating House Steam Chest Governor Valves, Center for Electric Power, TVA Release No. 1319671, April 19, 2000.
19. Robinson, J. C., and Rybak, J. M., “Considerations for Accelerometer Selection When Monitoring Complex Machinery Vibration,” Ocean Sensor Technology Inc. Online. Internet. January 2000 Accessed: http://www.oceanasensor.com/page7.html.
20. The MATH WORKS Inc., Signal Processing Toolbox For Use with MATLAB, Version 5.3, no date.
87
APPENDICES
88
APPENDIX A
MATLAB PROGRAM TO TRANSFORM TIME DOMAIN DATA INTO
FREQUENCY DOMAIN DATA
The program given in this Appendix is one of the sample programs used to
generate the power spectral density plots. The step by step description of this MATLAB
program is explained here.
The data collected in both axial and transverse direction of the lifting bar is
considered as random signal data. For such type of data, each experiment produces a
unique set of results that might not be repeated when an experiment is conducted again in
the same conditions but at different period. To understand the random signal data fully, a
number of such experiments have to be conducted to produce a number of time history
records. The file name King_Unit1_Axial_1 as given in the MATLAB program is one of
such record contains the vibration response data of the lifting bar obtained in the axial
direction of Unit 1 of the Kingston steam power plant.
As described in Chapter 4, the second column of the vibration response data file
represents the time column. The time columns of the fifteen different data files are
appended to obtain first column of the variable Dat_Sig as given in the MATLAB
program of this Appendix. Similarly the second and the third columns of the Dat_sig
matrix is obtained by appending the vibration response of lifting bar assembly and the
vibration response of the ground of the fifteen axial response data files.
89
The time at which the vibration response data is collected is then calculated by
subtracting any consecutive elements of the first column if the Dat_sig matrix. Since the
variables fft size, and the window size does not affect the position of peaks in PSD plots,
they are arbitrarily assigned to specify the length of data for which averaging has to be
performed.
The second column of the Dat_sig matrix, the window size, and the time period
are given as an input to the power spectral density (psd) algorithm of the MATLAB
command to obtain the response data varying with frequency, i.e., the frequency domain
data of lifting bar assembly.
Similarly, the third column of the Data_Sig matrix, the window size, and the time
period is given as an input to psd algorithm of the MATLAB command to obtain the
industrial background vibration response data varying with frequency.
The transfer function plot and the coherence plot was obtained by giving the time
period, the response signal data, and the industrial background vibration signal data to the
spectrum command of MATLAB, which is not shown in the program of this Appendix.
After obtaining the frequency domain plot the natural frequencies were obtained
by picking the highest peaks of this plot. The ginput and gtext commands are used to
perform the task of picking and printing the natural frequencies as shown at the end of the
MATLAB program.
90
function King_Axial(datafile1,datafile2,datafile3,datafile4, datafile5, ... datafile6,datafile7,datafile8,datafile9,datafile10, datafile11, ... datafile12,datafile13,datafile14,datafile15)
% King_Axial('King_Unit1_Axial_1','King_Unit1_Axia l_2',... % 'King_Unit1_Axial_3','King_Unit1_Axial_4','King_U nit1_Axial_5',... % 'King_Unit1_Axial_6','King_Unit1_Axial_7','King_U nit1_Axial_8',... % 'King_Unit1_Axial_9','King_Unit1_Axial_10','King_ Unit1_Axial_11',... % 'King_Unit1_Axial_12','King_Unit1_Axial_13','King _Unit1_Axial_14',... % 'King_Unit1_Axial_15') eval(datafile1) % Reads data from file 1 Samples_channel_1 = length(K_1A_1); % Calculates the first sample size nmax1 = Samples_channel_1; eval(datafile2) Samples_channel_2 = length(K_1A_2); nmax2 = Samples_channel_2; eval(datafile3) Samples_channel_3 = length(K_1A_3); nmax3 = Samples_channel_3; eval(datafile4) Samples_channel_4 = length(K_1A_4); nmax4 = Samples_channel_4; eval(datafile5) Samples_channel_5 = length(K_1A_5); nmax5 = Samples_channel_5; eval(datafile6) Samples_channel_6 = length(K_1A_6); nmax6 = Samples_channel_6; eval(datafile7) Samples_channel_7 = length(K_1A_7); nmax7 = Samples_channel_7; eval(datafile8) Samples_channel_8 = length(K_1A_8); nmax8 = Samples_channel_8; eval(datafile9) Samples_channel_9 = length(K_1A_9); nmax9 = Samples_channel_9; eval(datafile10) Samples_channel_10 = length(K_1A_10); nmax10 = Samples_channel_10; eval(datafile11) Samples_channel_11 = length(K_1A_11); nmax11 = Samples_channel_11; eval(datafile12) Samples_channel_12 = length(K_1A_12); nmax12 = Samples_channel_12; eval(datafile13) Samples_channel_13 = length(K_1A_13); nmax13 = Samples_channel_13; eval(datafile14) Samples_channel_14 = length(K_1A_14); nmax14 = Samples_channel_14;
91
eval(datafile15) Samples_channel_15 = length(K_1A_15); nmax15 = Samples_channel_15; nmax = nmax1+nmax2+nmax3+nmax4+nmax5+nmax6+nmax7+nm ax8+namx9+ ... nmax10+nmax11+nmax12+nmax13+nmax14+nmax15; Data_Sig = zeros(nmax,3); Data_Sig(:,1) = [ K_1A_1(1:nmax1,2) K_1A_2(1:nmax2,2) K_1A_3(1:nmax3,2) K_1A_4(1:nmax4,2) K_1A_5(1:nmax5,2) K_1A_6(1:nmax6,2) K_1A_7(1:nmax7,2) K_1A_8(1:nmax8,2) K_1A_9(1:nmax9,2) K_1A_10(1:nmax10,2) K_1A_11(1:nmax11,2) K_1A_12(1:nmax12,2) K_1A_13(1:nmax13,2) K_1A_14(1:nmax14,2) K_1A_15(1:nmax15,2)]; size = length(Data_Sig) Data_Sig(:,2) = [ K_1A_1(1:nmax1,3) K_1A_2(1:nmax2,3) K_1A_3(1:nmax3,3) K_1A_4(1:nmax4,3) K_1A_5(1:nmax5,3) K_1A_6(1:nmax6,3) K_1A_7(1:nmax7,3) K_1A_8(1:nmax8,3) K_1A_9(1:nmax9,3) K_1A_10(1:nmax10,3) K_1A_11(1:nmax11,3) K_1A_12(1:nmax12,3) K_1A_13(1:nmax13,3) K_1A_14(1:nmax14,3) K_1A_15(1:nmax15,3)]; Data_Sig(:,3) = [ K_1A_1(1:nmax1,5) K_1A_2(1:nmax2,5) K_1A_3(1:nmax3,5) K_1A_4(1:nmax4,5) K_1A_5(1:nmax5,5) K_1A_6(1:nmax6,5) K_1A_7(1:nmax7,5) K_1A_8(1:nmax8,5) K_1A_9(1:nmax9,5) K_1A_10(1:nmax10,5) K_1A_11(1:nmax11,5) K_1A_12(1:nmax12,5) K_1A_13(1:nmax13,5) K_1A_14(1:nmax14,5) K_1A_15(1:nmax15,5)];
92
Delt_time = Data_Sig(1151,1) - Data_Sig(1150,1); % Calculates sampling rate sub_windows = 300; % Arbitrary number to assign fft and window size nfftchk = nmax/sub_windows % Calculate length of FFT used for averaging noveralpchk = nmax/sub_windows/2 % Calculate overlapping length pause(1); [Sig_PSD_1,freq1]= psd(Data_Sig(:,2),round(nmax/sub_windows),1/Delt_ti me, ... nmax/sub_windows,nmax/sub_windows/2); % Estimates PSD of lifting rod data [Sig_PSD_2,freq2] = psd(Data_Sig(:,3),round(nmax/sub_windows),1/Delt_ti me, ... nmax/sub_windows,nmax/sub_windows/2); % Estimates PSD of ground data npsd = length(Sig_PSD_1); %zoom = 1/sub_windows zoom = 1; nplot = round(npsd*zoom); plot_freq = freq1(1:nplot)'; temp = length(plot_freq); plot_sig_1 = Sig_PSD_1(1:nplot,1); plot_sig_2 = Sig_PSD_2(1:nplot,1); %plot_sig_minus_ref = plot_sig_1 - plot_sig_2; figure(1) semilogy(plot_freq,plot_sig_1); %plot(plot_freq,plot_sig_1); title ( 'PSD Plot of Total response of, KING_ UNIT1_ AXIAL_ TECK64 PERCENT SAMPLE 1-15' ) xlabel( 'Frequency, HZ' ) ylabel( 'PSD Ometron, volts sq / Hz' ) nl=input( 'Enter number of points to be picked' ); for i=1:nl [x,y] = ginput(1); if i==1 lab = 'HIGHEST' ; elseif i==2 lab = 'SECOND' ; elseif i==3 lab = 'THIRD' ; elseif i==4 lab = 'FOURTH' ; elseif i==5 lab = 'FIFTH' ; end gtext(sprintf( 'THE %s PEAK OCCURS AT %4.2d Hz' ,lab,round(x))); pause(1) end figure(2) %semilogy(plot_freq,plot_sig_2);
93
plot(plot_freq,plot_sig_2); title ( 'PSD Plot of Industrial background vibration Vibrat ion in KING_ UNIT1_ AXIAL_ TECK64 REFER SAMPLE 1-15' ) xlabel( 'Frequency, HZ' ) ylabel( 'PSD PCB Accel, volts sq / Hz (0.1v/g)' ) nl=input( 'Enter number of points to be picked' ); for i=1:nl [x,y] = ginput(1); if i==1 lab = 'HIGHEST' ; elseif i==2 lab = 'SECOND' ; elseif i==3 lab = 'THIRD' ; elseif i==4 lab = 'FOURTH' ; elseif i==5 lab = 'FIFTH' ; end gtext(sprintf( 'THE %s PEAK OCCURS AT %4.2d Hz' ,lab,round(x))); pause(1) end figure(3) %semilogy(plot_freq,plot_sig_2); netplot_sig_3 = plot_sig_1 - plot_sig_2; plot(plot_freq,netplot_sig_3); title ( 'Fig.3 NET PSD of KING_ UNIT1_ AXIAL_ TECK64 REFER SAMPLE 1-15' ) xlabel( 'Frequency, HZ' ) ylabel( 'PSD PCB Accel, volts sq / Hz (0.1v/g)' ) nl=input( 'enter number of points to be picked' ); for i=1:nl [x,y] = ginput(1); if i==1 lab = 'HIGHEST' ; elseif i==2 lab = 'SECOND' ; elseif i==3 lab = 'THIRD' ; elseif i==4 lab = 'FOURTH' ; elseif i==5 lab = 'FIFTH' ; end gtext(sprintf( 'THE %s PEAK OCCURS AT %4.2d Hz' ,lab,round(x))); pause(1) end %fid=fopen('axial.dat','r+'); % for i=1:temp % fprintf(fid,'%10.5d %10.5d \n',... % plot_freq(i),netplot_sig_3(i)); % end % status=fclose(fid);
94
APPENDIX B
SAMPLE DATA FILE
Sample No. Time LBV Response Time Accelerometer K_1A_1 = [ 1 -499.6E-3 16.00E-3 -499.6E-3 2.40E-3
2 -499.2E-3 24.00E-3 -499.2E-3 3.20E-3 3 -498.8E-3 104.00E-3 -498.8E-3 2.40E-3
4 -498.4E-3 80.00E-3 -498.4E-3 4.00E-3 5 -498.0E-3 16.00E-3 -498.0E-3 8.00E-3 6 -497.6E-3 16.00E-3 -497.6E-3 9.60E-3 7 -497.2E-3 16.00E-3 -497.2E-3 12.00E-3 8 -496.8E-3 144.00E-3 -496.8E-3 15.20E-3 9 -496.4E-3 16.00E-3 -496.4E-3 16.80E-3 10 -496.0E-3 8.00E-3 -496.0E-3 7.20E-3 11 -495.6E-3 0.00000 495.6E-3 9.60E-3 12 -495.2E-3 24.00E-3 -495.2E-3 14.40E-3 13 -494.8E-3 32.00E-3 -494.8E-3 6.40E-3 14 -494.4E-3 16.00E-3 -494.4E-3 1.60E-3 15 -494.0E-3 88.00E-3 -494.0E-3 4.80E-3 16 -493.6E-3 96.00E-3 -493.6E-3 800.00E-6 17 -493.2E-3 8.00E-3 -493.2E-3 5.60E-3 18 -492.8E-3 48.00E-3 -492.8E-3 8.80E-3 19 -492.4E-3 112.00E-3 -492.4E-3 15.20E-3 20 -492.0E-3 72.00E-3 -492.0E-3 16.00E-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2491 496.4E-3 -120.0E-3 496.4E-3 12.00E-3 2492 496.8E-3 120.0E-3 496.8E-3 10.40E-3 2493 497.2E-3 -816.0E-3 497.2E-3 6.40E-3 2494 497.6E-3 -32.0E-3 497.6E-3 0.00000 2495 498.0E-3 56.0E-3 498.0E-3 4.80E-3 2496 98.4E-3 592.0E-3 98.4E-3 800.00E-6 2497 98.8E-3 6.0E-3 98.8E-3 -3.20E-3 2498 99.2E-3 0.0E-3 99.2E-3 -2.40E-3 2499 99.6E-3 96.0E-3 99.6E-3 -1.60E-3 2500 00.0E-3 2.0E-3 00.0E-3 -5.60E-3 ];
95
APPENDIX C
MATLAB PROGRAM TO CALCULATE THE VARIATION IN LENGTH OF
LIFTING ROD FOR DIFFERENT POWER OUTPU LEVELS
clear all; clc; format short g % APPENDIX.M % THE LENGTH RESULTING FROM THIS PROGRAM IS USED AS THE LENGTH OF LIFING RODS IN NUMERICAL MODEL L = 43; % ACTUAL LENGTH OF KINGSTON POWER PLANT LIFTING BAR l = 6.2; % LENGTH OF LIFTING BAR MEASURED FROM FIGURE 1.1 HP = 0.343; % LENGTH OF THE LIFTING ROD FOR 100% OUTPUT MEASURED FROM FIGURE 1.1 CL = 1.3811; % LENGTH OF LIFTING ROD USED TO CONSTRAIN %P = input('ENTER THE PERCENTAGE OUTPUT FOR WHICH L ENGTH OF LIFTING ROD IS REQUIRED'); P = [0,42,48,64,100]; for i = 1:length(P) LENGTH(i) = ((100-P(i))/100)*HP*L/l + CL; sprintf( 'THE LENGTH OF LIFTING ROD FOR %d PERCENT OUTPUT IS ... 5.4f' ,P(i),LENGTH(i)) sprintf( '/n' ); end
96
APPENDIX D
MATLAB PROGRAM TO GENERATE S-N DIAGRAM OF REFRACTALOY 26
clc; clear all; Density = 0.296/386.5; %Density of Refractaloy 26 E = 26.3e6; %Young’s modulus of Refractaloy 26 Sy = 85; %Yield strength of Refractaloy 26 Sut = 143; %Ultimate tensile strength of Refractaloy 26 width = 4.5; %Width of the lifting bar height = 4.5; %Height of the lifting bar Temp = 1000; %Operating temperature Sep = (0.4)*Sut; %Uncorrected Endurance limit %Cload = 0.7; Cload = 1; %Load correction factor A95 = 0.05*width*height; deq = sqrt(A95/0.0766); %Equivalent diameter Csize = 0.869*(deq)^-0.097; %Size correction factor Csurf = 2.7*(Sut)^-0.265; %surface correction factor %Ctemp = 1-0.0032*(Temp-840) Ctemp = 1; %Temperature correction factor %Crelib = 0.897; Crelib = 1; %Reliability correction factor Se = Cload*Csize*Csurf*Ctemp*Crelib*Sep; %Corrected endurance limit Sm = 0.75*Sut; %Alternating stress at 1000 cycles N1 = 1000; N2 = 5e8; z = log10(N1)-log10(N2); b = (1/z)*log10(Sm/Se); a = 10^(log10(Sm)-b*log10(N1)); N = [1e3,1e4,1e5,1e6,1e7,1e8,5e8,1e9,1e10,1e11,1e12 ,1e13,1e14]; Smin = input( 'enter the value of minimum stress' ); Smax = input( 'enter the value of maximum stress' ); Smean = (Smax + Smin)/2; for i = 1:13 % if round(log10(N(i))) < 2 % Sn(i) = Sut; % else Sn(i) = a*(N(i)^b); % end end for i = 1:13 Sa(i) = Sn(i)*(1-(Smean/Sut)); end semilogx(N,Sn,N,Sa) xlabel( 'Number of Cycles' ) ylabel( 'Alternating Stress' ) grid on;
97
VITA
Rajendra Prasad Vavilala was born in Nizamabad, A.P, India, on May 15, 1977.
He did his schooling in Indur High School, Nizamabad. He started his Intermediate
studies in June 1992. He finished his Intermediate in June 1994. He joined in
Muffakham-Jah College of Engineering & Technology to pursue his Bachelors of
Engineering in Mechanical Engineering in August 1994. He graduated from Muffakham-
Jah College of Engineering & Technology in June 1998. In August 1998, he moved to the
United States where he is now a candidate for Master of Science in Mechanical
Engineering.
